Medical Physics - Gases.pdf

Medical Physics Gases Ideal gas Medical Physics| Gases Contents : Ideal gas 3 Properties of gases 7 Discovering the Relationships Between Properties 11 Explanations for Other Real-World Situations 30 Ideal gas calculations 33 Medical Physics| Gases Ideal gas : If you want to understand how gases beh...

Medical Physics Gases Ideal gas Medical Physics| Gases Contents : Ideal gas 3 Properties of gases 7 Discovering the Relationships Between Properties 11 Explanations for Other Real-World Situations 30 Ideal gas calculations 33 Medical Physics| Gases Ideal gas : If you want to understand how gases behave, such as why fresh air rushes into your lungs when certain chest muscles contract or how gases in a car’s engine move the pistons and power the car, you need a clear mental image of the model chemists use to explain the properties of gases and the relationships between them. Gases consist of tiny particles widely spaced. Under typical conditions, the average distance between gas particles is about ten times their diameter. Because of these large distances, the volume occupied by the particles themselves is very small compared to the volume of the empty space around them. Medical Physics| Gases For a gas at room temperature and pressure, the gas particles themselves occupy about 0.1% of the total volume. The other 99.9% of the total volume is empty space (whereas in liquids and solids, about 70% of the volume is occupied by particles). Because of the large distances between gas particles, the attractions or repulsions among them are weak. Medical Physics| Gases The model described above applies to real gases, but chemists often simplify the model further by imagining the behavior of an ideal gas. Medical Physics| Gases Ideal Gases : An ideal gas differs from a real gas in that: The particles are assumed to be point masses, that is, particles that have a mass but occupy no volume. There are no attractive or repulsive forcesat all between the particles. Medical Physics| Gases Properties of Gases : The ideal gas model is used to predict changes in four related gas properties: volume, number of particles, temperature, and pressure. Volumes of gases are usually described in liters L, or cubic meters m3, and numbers of particles are usually described in moles mol. Although gas temperatures are often measured with thermometers that report temperatures in degrees Celsius °C, scientists generally use Kelvin temperatures for calculations. Medical Physics| Gases Remember that you can convert between degrees Celsius °C and kelvins K. To understand gas pressure, imagine a typical gas in a closed container. Each time a gas particle collides with the walls of its container, it exerts a force against the wall. The sum of the forces of these ongoing collisions of gas particles against all the container’s interior walls creates a continuous pressure upon those walls. Pressure is force divided by area. Medical Physics| Gases The accepted SI unit for gas pressure is the Pascal Pa. A Pascal is a very small amount of pressure, so the kilopascal kPa is more commonly used. Other units used to describe gas pressure are the atmosphere (atm), torr, millimeter of mercury (mmHg), and bar. The relationships between these pressure units are: 1 atm = 101,325 Pa = 101.325 kPa = 760 mmHg = 760 torr 1 bar = 100 kPa = 0.9869 atm = 750.1 mmHg Medical Physics| Gases The numbers in these relationships come from definitions, so they are all exact. At sea level on a typical day, the atmospheric pressure is about 101 kPa, or about 1 atm. In calculations, the variables P, T, V, and n are commonly used to represent pressure, temperature, volume, and moles of gas. Medical Physics| Gases Discovering the Relationships Between Properties : A simple system is considered in which two of the four gas properties are held constant, a third property is varied, and the effect of this variation on the fourth property is observed. For example, it is easier to understand the relationship between volume and pressure if the number of gas particles and temperature are held constant. The volume can be varied, and the effect this has on the pressure can be measured. An understanding of the relationships between gas properties in controlled situations will help us to explain and predict the effects of changing gas properties in more complicated, real situations. Medical Physics| Gases The Relationship Between Volume and Pressure : While holding the number of gas particles constant (by closing the valve) and holding the temperature constant (by allowing heat to transfer in or out so that the apparatus remains the same temperature as the surrounding environment), we move the piston to change the volume, and then we observe the change in pressure. When we decrease the gas volume, the pressure gauge on our system shows us that the gas pressure increases. When we increase the gas volume, the gauge shows that the pressure goes down. Medical Physics| Gases For an ideal gas (in which the particles occupy no volume and experience no attractions or repulsions), gas pressure and volume are inversely proportional. This means that if the temperature and the number of gas particles are constant and if the volume is decreased to one-half its original value, the pressure of the gas will double. If the volume is doubled, the pressure decreases to one-half its original value. The following expression summarizes this inverse relationship: Medical Physics| Gases Real gases deviate somewhat from this mathematical relationship, but the general trend of increased pressure with decreased volume (or decreased pressure with increased volume) is true for any gas. The observation that the pressure of an ideal gas is inversely proportional to the volume it occupies if the number of gas particles and the temperature are constant is a statement of Boyle’s Law. Medical Physics| Gases This relationship can be explained in the following way: When the volume of the chamber decreases but the number of gas particles remains constant, there is an increase in the concentration (number of particles per liter) of the gas. This leads to an increase in the number of particles near any given area of the container walls at any time and to an increase in the number of collisions against the walls per unit area in a given time. More collisions mean an increase in the force per unit area, or pressure, of the gas. Medical Physics| Gases The logic sequence presented in Figure summarizes this explanation. The arrows in the logic sequence can be read as “leads to.” Take the time to read the sequence carefully to confirm that each phrase leads logically to the next. Medical Physics| Gases The Relationship Between Pressure and Temperature : In order to examine the relationship between pressure and temperature, we must adjust our demonstration apparatus so that the other two properties (number of gas particles and volume) are held constant. This can be done by locking the piston so it cannot move and closing the valve tightly so that no gas leaks in or out. When the temperature of a gas trapped inside the chamber is increased, the measured pressure increases. When the temperature is decreased, the pressure decreases. Medical Physics| Gases We can explain the relationship between temperature and pressure using our model for gas. Increased temperature means increased motion of the particles. If the particles are moving faster in the container, they will collide with the walls more often and with greater force per collision. This leads to a greater overall force pushing on the walls and to a greater force per unit area or pressure. If the gas is behaving like an ideal gas, a doubling of the Kelvin temperature doubles the pressure. If the temperature decreases to 50% of the original Kelvin temperature, the pressure decreases to 50% of the original pressure. Medical Physics| Gases This relationship can be expressed by saying that the pressure of an ideal gas is directly proportional to the Kelvin temperature of the gas if the volume and the number of gas particles are constant. This relationship is sometimes called Gay-Lussac’s Law. Medical Physics| Gases The Relationship Between Volume and Temperature : To demonstrate the relationship between temperature and volume of gas,we keep the number of gas particles and gas pressure constant. If our valve is closed and if our system has no leaks, the number of particles is constant. We keep the gas pressure constant by allowing the piston to move freely throughout our experiment, because then it will adjust to keep the pressure pushing on it from the inside equal to the constant external pressure pushing on it due to the weight of the piston and the atmospheric pressure. Medical Physics| Gases The atmospheric pressure is the pressure in the air outside the container, which acts on the top of the piston due to the force of collisions between particles in the air and the top of the piston. If we increase the temperature, the piston in our apparatus moves up, increasing the volume occupied by the gas. A decrease in temperature leads to a decrease in volume. Medical Physics| Gases The increase in temperature of the gas leads to an increase in the average velocity of the gas particles, which leads in turn to more collisions with the walls of the container and a greater force per collision. This greater force acting on the walls of the container leads to an initial increase in the gas pressure. Thus, the increased temperature of our gas creates an internal pressure, acting on the bottom of the piston, that is greater than the external pressure. The greater internal pressure causes the piston to move up, increasing the volume of the chamber. The increased volume leads to a decrease in gas pressure in the container, until the internal pressure is once again equal to the constant external pressure. Medical Physics| Gases Similar reasoning can be used to explain why decreased temperature leads to decreased volume when the number of gas particles and pressure are held constant. For an ideal gas, volume and temperature described in kelvins are directly proportional if the number of gas particles and pressure are constant. This is called Charles’ Law. Medical Physics| Gases The Relationship Between Number of Gas Particles and Pressure : The volume is held constant by locking the piston so it cannot move. The temperature is kept constant by allowing heat to flow in or out of the cylinder in order to keep the temperature of the gas in the cylinder equal to the external temperature. When the number of gas particles is increased by adding gas through the valve on the left of the cylinder, the pressure gauge shows an increase in pressure. When gas is allowed to escape from the valve, the decrease in the number of gas particles causes a decrease in the pressure of the gas. Medical Physics| Gases Increased number of gas particles → Increased pressure. Decreased number of gas particles → Decreased pressure. The increase in the number of gas particles in the container leads to an increase in the number of collisions with the walls per unit time. This leads to an increase in the force per unit area–that is, to an increase in gas pressure. Medical Physics| Gases If the temperature and the volume of an ideal gas are held constant, the number of gas particles in a container and the gas pressure are directly proportional. Medical Physics| Gases The Relationship Between Number of Gas Particles and Volume : Temperature is held constant by allowing heat to move into or out of the system, thus keeping the internal temperature equal to the constant external temperature. Pressure is held constant by allowing the piston to move freely to keep the internal pressure equal to the external pressure. When we increase the number of gas particles in the cylinder by adding gas through the valve on the left of the apparatus, the piston rises, increasing the volume available to the gas. If the gas is allowed to escape from the valve, the volume decreases again. Medical Physics| Gases The explanation for why an increase in the number of gas particles increases volume starts with the recognition that the increase in the number of gas particles results in more collisions per second against the walls of the container. The greater force due to these collisions creates an initial increase in the force per unit area—or gas pressure—acting on the walls. This will cause the piston to rise, increasing the gas volume and decreasing the pressure until the internal and external pressure are once again equal. Medical Physics| Gases The relationship between moles of an ideal gas and volume is summarized by Avogadro’s Law, which states that the volume and the number of gas particles are directly proportional if the temperature and pressure are constant. Medical Physics| Gases Explanations for Other Real-World Situations : The relationships between gas properties can be used to explain how we breathe in and out. When the muscles of your diaphragm contract, your chest expands, and the volume of your lungs increases. This change leads to a decrease in the number of particles per unit volume inside the lungs, leaving fewer particles near any give area of the inner surface of the lungs. Fewer particles mean fewer collisions per second per unit area of lungs and a decrease in force per unit area, or gas pressure. Medical Physics| Gases During quiet, normal breathing, this increase in volume decreases the pressure in the lungs to about 0.4 kilopascals lower than the atmospheric pressure. As a result, air moves into the lungs faster than it moves out, bringing in fresh oxygen. When the muscles relax, the lungs return to their original, smaller volume, causing the pressure in the lungs to increase to about 0.4 kilopascals above atmospheric pressure. Air now goes out of the lungs faster than it comes in. Medical Physics| Gases Medical Physics| Gases Ideal Gas Calculations : We know that pressure of an ideal gas is directly proportional to the number of gas particles (expressed in moles), directly proportional to temperature, and inversely proportional to the volume of the container. Medical Physics| Gases These three relationships can be summarized in a single equation: Another way to express the same relationship is The constant in this equation is the same for all ideal gases. It is called the universal gas constant and is expressed with the symbol R. The value of R depends on the units of measure one wishes to use in a given calculation. Two choices showing R for different pressure units (atmospheres, atm, and kilopascals, kPa). Medical Physics| Gases Substituting R for “a constant” and rearranging the equation yields the ideal gas equation (often called the ideal gas law) in the form that is most commonly memorized and written: PV = nRT Because we will often be interested in the masses of gas samples, it is useful to remember an expanded form of the ideal gas equation that uses mass in grams (g) divided by molar mass (M ) instead of moles (n). Medical Physics| Gases When Properties Change : Another useful equation, derived from the ideal gas equation, can be used to calculate changes in the properties of a gas. In the first step of the derivation, the ideal gas equation is rearranged to isolate R. In this form, the equation shows that no matter how the pressure, volume, moles of gas, or temperature of an ideal gas may change, the other properties will adjust so that the ratio of PV/nT always remains the same. Medical Physics| Gases In the equations below, P1, V1, n1, and T1 are the initial pressure, volume, moles of gas, and temperature, and P2, V2, n2, and T2 are the final pressure, volume, moles of gas, and temperature. Medical Physics| Gases Example 1: Neon gas in luminous tubes radiates red light—the original “neon light.” The standard gas containers used to fill the tubes have a volume of 1.0 L and store neon gas at a pressure of 101 kPa at 22 °C. A typical luminous neon tube contains enough neon gas to exert a pressure of 1.3 kPa at 19 °C. If all the gas from a standard container is allowed to expand until it exerts a pressure of 1.3 kPa at 19 °C, what will its final volume be? Medical Physics| Gases Solution V1 = 1.0 L P1 = 101 kPa T1 = 22 °C + 273.15 = 295 K V2 = ? P2 = 1.3 kPa T2 = 19 °C + 273.15 = 292 K We write the combined gas law equation, eliminating variables for properties that are constant. Because moles of gas are not mentioned, we assume that they are constant (n1 = n2). Medical Physics| Gases Example 2: A balloon with a volume of 2.0 L is filled with a gas at 3 atmospheres. If the pressure is reduced to 0.5 atmospheres without a change in temperature, what would be the volume of the balloon? Medical Physics| Gases Solution Since the temperature doesn't change, Boyle's law can be used. Boyle's gas law can be expressed as: PiVi = PfVf Pi = initial pressure Vi = initial volume Pf = final pressure Vf = final volume Medical Physics| Gases Solution To find the final volume, solve the equation for Vf: Vf = PiVi/Pf Vf = (2.0 L) (3 atm) / (0.5 atm) Vf = 6 L / 0.5 atm Vf = 12 L The volume of the balloon will expand to 12 L. Medical Physics| Gases Example 3: Calculate the decrease in temperature (in Celsius) when 2.00 L at 21.0 °C is compressed to 1.00 L. Solution