An ideal gas expands isothermally at a temperature of 300 K from an initial volume of 2 L to a final volume of 5 L. If the pressure at the initial state is 4 atm, calculate the wor... An ideal gas expands isothermally at a temperature of 300 K from an initial volume of 2 L to a final volume of 5 L. If the pressure at the initial state is 4 atm, calculate the work done by the gas.
Understand the Problem
The question is asking us to calculate the work done by an ideal gas during an isothermal expansion from an initial volume to a final volume at a given temperature, using the initial pressure of the gas. We'll use the formula for work done in isothermal processes in ideal gases.
Answer
The work done by the ideal gas during an isothermal expansion is given by the formula \( W = nRT \ln\left(\frac{V_f}{V_i}\right) \).
Answer for screen readers
The work done by the ideal gas during an isothermal expansion is given by:
$$ W = nRT \ln\left(\frac{V_f}{V_i}\right) $$
Steps to Solve
- Identify the relevant formula
For an ideal gas undergoing isothermal expansion, the work done, ( W ), can be calculated using the formula:
$$ W = nRT \ln\left(\frac{V_f}{V_i}\right) $$
where ( n ) is the number of moles of gas, ( R ) is the ideal gas constant, ( T ) is the absolute temperature, ( V_f ) is the final volume, and ( V_i ) is the initial volume.
- Gather the necessary values
You need:
- ( n ): number of moles (given or calculated)
- ( R ): the ideal gas constant, ( R = 8.314 , \text{J/(mol K)} )
- ( T ): absolute temperature in Kelvin
- ( V_f ): final volume
- ( V_i ): initial volume
Make sure all units are consistent.
- Calculate the natural logarithm of the volume ratio
Compute the ratio of volumes:
$$ \frac{V_f}{V_i} $$
Then, find the natural logarithm:
$$ \ln\left(\frac{V_f}{V_i}\right) $$
- Substitute values into the formula
Insert your gathered values into the work done formula:
$$ W = nRT \ln\left(\frac{V_f}{V_i}\right) $$
Calculate this expression to find the work done.
- Interpret the result
Analyze the calculated work value to understand the significance of your results in context of the gas behavior during the isothermal expansion.
The work done by the ideal gas during an isothermal expansion is given by:
$$ W = nRT \ln\left(\frac{V_f}{V_i}\right) $$
More Information
In an isothermal process, the temperature remains constant, leading to no change in internal energy for an ideal gas. Hence, the work done is equal to the heat absorbed by the gas. This formula illustrates the link between temperature, volume change, and work done.
Tips
- Not converting all temperatures to Kelvin before calculation.
- Forgetting to use the natural logarithm when calculating the volume ratio.
- Using incorrect units for pressure, volume, or the gas constant that can lead to incorrect results.
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