TP de Thermodynamique n°1: Calorimétrie PDF

# TP de thermodynamique n°1: Calorimétrie ## I. But The goal of this experiment is to determine the value of J, the coefficient relating the joule and the calorie, using the electrical method. ## II. Theoretical part Calorimetry involves experimentally determining quantities of heat. Heat, a fo...

# TP de thermodynamique n°1: Calorimétrie ## I. But The goal of this experiment is to determine the value of J, the coefficient relating the joule and the calorie, using the electrical method. ## II. Theoretical part Calorimetry involves experimentally determining quantities of heat. Heat, a form of energy, is a physical quantity which is not directly measurable; so we must measure its effect on another body. These effects can be: * Used to increase or decrease the temperature of a body without changing its physical state; this refers to the concept of specific heat. * Used to cause a change in the physical state of a body without modifying its temperature; this refers to the concept of latent heat. The specific heat of a body is the quantity of heat needed for one unit of mass of that body to increase its temperature by one degree. Latent heat is the heat necessary to cause a change in the physical state (vaporization, fusion, etc.) of one unit of mass of a body at constant temperature. Specific and latent heat, as well as other thermal quantities, can be measured using a device called a calorimeter. This device is made of a vessel, called a calorimetric vessel, which is thermally isolated from the outside environment. J. P. JOULE (1818-1889) demonstrated that heat and work are two modes of energy transfer (the 1st principle of thermodynamics). Several modes of heat transfer can be considered: * Any body at a temperature emits thermal radiation in the form of electromagnetic waves. This refers to the transfer of heat by radiation. * When two bodies A and B are in contact with a fluid (e.g., Air), this fluid heats up when in contact with a hot body, then cools down when in contact with a cold body, and transfers heat to it. This refers to the transfer of heat by convection. * If two points of the same body are at different temperatures, the temperature varies continuously until all points of the same body are at the same temperature. This refers to the transfer of heat by conduction ### 1. Differential form of heat During an elementary transformation, the quantity of heat exchanged between a system and its external environment (see the course) can be written using the (P,T) diagram: $\delta Q = C_p \ dT + h \ dP$ (1) where $C_p$ is the specific heat capacity at constant pressure, and _h_ is a calorimetric coefficient. When the transformation occurs at constant pressure, which will be the case during this experiment, equation (1) becomes: $\delta Q = C_p \ dT$ (2) The heat capacity per unit of mass, corresponding to the specific heat capacity, is defined as $\Large{c}_p$. We can write equation (2) as: $\delta Q = m \ c_p \ dT$ (3) ### 2. Electrical method The electrical method involves heating the water by the Joule effect. The passage of a current I(A) through a resistance, immersed in a mass of water m(g), will increase the temperature of the water from T_i (° C) to T_f (° C) in time t(s). The energy supplied by the Joule effect is: W = U.I.t (in joules) Assuming no heat loss, the heat received by the water mass m is obtained from equation (3): Q = m.c_p (T_f - T_i) (in calories) In reality, we need to take into account the heat absorbed by the calorimeter's walls. This heat loss is equivalent to the heat absorbed by a mass of water _µ_ (water equivalent of the calorimeter). Q_µ = µ.c_p (T_f - T_i) The total heat received by the water and the calorimeter is: Q = Q_m + Q_µ = (m+ µ) c_p (T_f - T_i) The coefficient J relating the quantity of heat Q expressed in calories to the work W expressed in joules is obtained from: J = (m+ µ) c_p (T_f - T_i) / W = (m + µ) c_p (T_f- T_i) / U.I.t where c_p = (1 ± 0.1) cal/g°C and µ = (60 ± 5)g. A correction on the term (T_f - T_i) can be made. To do this, plot the curve T = f(t) which allows to take into account the heat loss (see figure 5). If we consider that heating occurs instantly at the average time (t'+t") / 2, then the temperature variation is determined graphically: (T_f -T_i)_corrected = (T" -T_B' ) **Remarks:** The equilibrium state of our system is defined as follows: * **Thermal equilibrium:** The temperature is homogeneous throughout the system (hence the use of a magnetic stirrer) * **Mechanical equilibrium:** At the water surface, a mechanical equilibrium exists since the two forces of pressure are equal; this allows us to consider that the transformation occurs at constant pressure. ## III. Equipment The calorimeter used in this experiment is the Berthelot calorimeter. It is essentially made up of two cylindrical aluminium containers, one containing water, the other empty. These containers form an adiabatic enclosure (no heat exchange with the outside environment). The larger container only contains the smaller container (calorimetric vessel). The latter contains the water, the magnetic stir bar and the accessories (Figure 1). [Diagram of the Calorimeter] The calorimeter is closed by a lid with holes allowing the passage of a thermometer and an electrical resistance heater. The experimental device includes a generator, a magnetic stirrer, a thermometer, a stopwatch, electrical connections, an ammeter, a voltmeter and a stopwatch (Figure 2). [Diagram of the Experimental Setup] ## IV. Experimental procedure The experiment consists of determining the coefficient J relating the heat to the work by monitoring the change in the temperature T of a given mass of water m as a function of time t. To do this, follow the steps below: * Remove the calorimetric vessel from the adiabatic enclosure. * Remove the lid from the vessel. * Weigh the empty calorimetric vessel using the balance. Note its mass. * Fill the vessel with a mass m of cold water (the value of m will be given to you in the room). * Secure the lid to the calorimetric vessel filled with water, then place the assembly in the empty adiabatic enclosure (Figure 3). * Place the resistance heater and the thermometer in the calorimetric vessel filled with water. * Set up the assembly as in Figure 4, and have it checked by your teacher. **Caution:** Do not turn on the generator. * Turn on the magnetic stirrer that rotates the magnetic stir bar. The rotation rate should be low. The magnetic stirrer must stay on throughout the experiment. * Turn on the stopwatch. It should only be turned off at the end of the experiment. * Turn on the generator. * Allow the generator to warm up without turning on the heater. Record the temperature of the water at times t = 0 minutes. * At t = 2 minutes, record the temperature of the water and turn on the heater. (see Figure 5). * Measure the voltage U and the current I using the voltmeter and the ammeter. * Record the temperature every minute until the ninth minute. * At the tenth minute, record the temperature and turn off the generator. The heating time in this example is 8 minutes. Continue to record the temperature until the 14th minute. * Turn off the stopwatch and the magnetic stirrer. **Remarks:** * The temperature measurements are taken at the beginning of each minute. * The values of t are given here as examples: they may be changed by your teacher. * Provide detailed calculations. ### Work to be done in the report: 1. Fill in the tables: **Table 1: Cold Water + Stirring + Generator off** | t (min) | T (°C) | |---|---| | 0 | | | 1 | | **Table 2: Water + Stirring + Generator on** | t (min) | T (°C) | |---|---| | 2 | | | 3 | | | 4 | | | 5 | | | 6 | | | 7 | | | 8 | | | 9 | | **Table 3: Hot Water + Stirring + Generator off** | t (min) | T (°C) | |---|---| | 10 | | | 11 | | | 12 | | | 13 | | | 14 | | 2. Plot the curve T = f(t) on graph paper. 3. Fill in the following table (t is the heating time): | U(V) | ΔU (V) | I(A) | ΔI(A)| m(g) | Δm(g) | t(s) | Δt(s) | T_B (°C) | ΔT_B (°C) | T_C (°C) | ΔT_C (°C) | |---|---|---|---|---|---|---|---|---|---|---|---| ||| **Note:** For the measurement of voltage U and current I as well as their uncertainties, see the section "USE OF THE AMMETER AND VOLTMETER" on page 10. The ammeter range is 10 A. 4. Give the numerical value of J + ΔJ, knowing that: J = (m+ µ) c_p (T_f -T_i) / UIt ΔJ = J (ΔU/U + ΔI/I + Δt/t + Δc_p/c_p + Δm/(m+µ) + ΔT_f / (T_f -T_i)) 5. Compare this value of J with the one obtained using a more sophisticated calorimeter than the Berthelot calorimeter (4.18 ± 0.01) J/cal. Draw conclusions. ## 3. Construction of a graph ### a. Plotting the experimental points M_i(r_i, y_i) on a graph Once the experimental measurements are made, a table of results is generated which contains the values of the variables r_i and y_i determined experimentally, as well as their respective absolute uncertainties Δr_i and Δy_i. | r_i | Δr_i | y_i | Δy_i | |---|---|---|---| | | | | | | | | | | | | | | | | | | | | The graphical representation of experimental results is done on graph paper. Start by selecting scales along the two axes. For example: * 1 unit of r = X mm * 1 unit of y = Y mm Two fundamental rules must be followed when choosing the scales and the origin: * Use as much space as possible. * Make the graph easy to read (simple scales). Once these considerations are made, plot the points M_i(x_i, y_i) on the graph using the selected scales. ### b. Plotting the error rectangles. By associating X_mm with the unit of variable r_i, you can determine the number of millimeters a_i associated with the values of the absolute uncertainties Δr_i. In the same way, associate the number of millimeters b_i with the values of the absolute uncertainties Δy_i. In practice, this involves a simple rule of three: * 1 unit of r = X_mm * 1 unit of y = Y_mm * a_i = Δr_i * X_mm * b_i = Δy_i * Y_mm Each point M_i(r_i, y_i) is associated with an error rectangle centered on this point and which has sides 2Δr_i and 2Δy_i (r_i - Δr_i ≤ r_i ≤ r_i + Δr_i and y_i - Δy_i ≤ y_i ≤ y_i + Δy_i). **Notes:** * If Δr_i and Δy_i are negligible, the values calculated for a_i or b_i are very small (less than 1/2 millimeter). The error rectangle is reduced to an error bar (or error segment), or to a single point, as illustrated below: [Diagram of the error rectangles] * In general, the graphs obtained after experimental measurements are displayed as shown in Figure 2. ### 3. Graph analysis #### a. Definitions Two important characteristics of the straight line y = ax + b are its slope a (also denoted by the symbol _p_) and its y-intercept b. **Slope of the straight line:** It is often denoted by the letter _p_. It is defined by: p = tg(θ) = (y_2 - y_1) / (x_2 -x_1) where M_1(x_1, y_1) and M_2(x_2, y_2) represent two points on the straight line (Figure 3). **Note:** For a given straight line, the slope is independent of the selected points. **Y-intercept:** Often denoted with the symbol _b_, is the y-coordinate of the point where the straight line intersects the y-axis (Figure 3). The equation of the straight line can then be written as: y = px + b where p is the slope of the straight line and _b_ is its y-intercept. #### b. Graphical determination of the slope and y-intercept When all the error rectangles are plotted on the graph, we can plot the minimum slope and the maximum slope (extreme slopes): * **Minimum slope p_min:** It corresponds to the smallest angle θ.. It is represented in Figure 4 by the straight line p_min passing through all the error rectangles and corresponding to the smallest angle θ. * **Maximum slope p_max:** It corresponds to the largest angle θ. It is represented in Figure 4 by the straight line p_max passing through all the error rectangles and corresponding to the largest angle θ. [Diagram of the graphs of slopes] The average slope p_moy and its uncertainty Δp_moy are given by: p_moy = (p_max + p_min) / 2 Δp_moy = (p_max - p_min) / 2 The straight lines (l_min) and (l_max) intersect the y-axis at points (0, b_min) and (0, b_max), respectively (Figure 1). The average value b_moy and its uncertainty Δb_moy are then: b_moy = (b_max + b_min)/2 Δb_moy = (b_max - b_min)/2 **Special case: Case where b = 0 (affine function):** This is the case where the variables r and y measured are linked by a relationship of the type y = ar. **Important note:** The third point is an outlier (Figure 6): An outlier is a point that was not measured correctly. In this case, it is necessary to repeat the measurement and plot the corresponding point on the graph. In Figure 7, the straight line (l_min) is wrongly plotted because it does not pass through the error rectangle of the fourth point. [Diagram of outliers] Pr A.Hamine