Collection of Solved Problems in Physics

Work, Pressure and Heat of the Air during Isothermal Expansion

Task number: 2179

• Notation

  V1 = 1 m3	initial volume of the gas
  p1 = 200 kPa = 2·105 Pa	initial pressure of the gas
  V2 = 2V1	volume of the gas after expansion
  W = ?	work performed by the gas
  p2 = ?	final pressure
  Q = ?	supplied heat

• Hint 1: Work W

  Think about how to calculate the work performed by the air if pressure is a function of volume.

• Solution of Hint 1

  Since the pressure p in this task is not constant, yet it is the function of volume, i.e. p(V), integration will be necessary to calculate the work. Thus, we use the following formula for the performed work

  \[W = \int\limits_{V_1}^{V_2}p(V)\, \text{d}V,\]

  where V₁ stands for initial volume and V₂ for final volume of the air.

• Hint 2: Expressing Pressure p(V)

  To express pressure p as a function of volume V, use the so-called Boyle-Mariotte law.

• Boyle-Mariotte Law

  According to the Boyle-Mariotte law:

  \[pV=const,\]

  where p stands for pressure and V for volume of a gas in an isothermal process.

• Hint 3: Resultant Pressure

  To determine the resultant pressure of the air, use the Boyle-Mariotte law again.

• Hint 4: Supplied Heat

  Internal energy U of the gas remains unchanged during an isothermal process, therefore all heat Q received by the system will be converted into work W performed by the gas.

• Analysis

  To calculate the work done by the air, we will have to use integration, because pressure is the function of volume.

  First, we express pressure of the air from the so-called Boyle-Mariotte law, which applies to isothermal processes, and integrate the obtained function with respect to volume. Initial and final volume of the air serve as the limits of the integral.

  Then we use the Boyle-Mariotte law again to calculate the resultant pressure of the air.

  In an isothermal process, heat received by the gas is equal to the work it performs (internal energy does not change).

• Solution

  To the work W performed by the air during isothermal expansion the general formula can be applied:

  \[W = \int\limits_{V_1}^{V_2}p(V)\, \text{d}V,\]

  where p stands for pressure, V₁ and V₂ for initial and final volume of the air respectively.

  Thus, we need to express pressure p as a function of volume V first. To do so, we use the Boyle-Mariotte law, according to which:

  \[p_1V_1 = pV. \]

  Therefrom we easily express pressure p:

  \[p = \frac{p_1V_1}{V}.\]

  Now we can proceed to the integration itself

  \[W = \int\limits_{V_1}^{V_2}p\, \text{d}V = \int\limits_{V_1}^{2V_1}\frac{p_1V_1}{V}\, \text{d}V =\]

  we factor the constants out of the integral

  \[=p_1V_1 \int\limits_{V_1}^{2V_1}\frac{1}{V}\, \text{d}V = \]

  integrate and substitute the limits

  \[=p_1V_1[\ln\,V]_{V_1}^{2V_1} = p_1V_1\,\ln \frac{2V_1}{V_1}= p_1V_1\,\ln\,2.\]

  The resultant pressure p₂ of the air can be determined from the above-mentioned Boyle-Mariotte law as well

  \[p_1V_1=p_2V_2.\]

  Therefrom we express the final pressure

  \[p_2=\frac{p_1V_1}{V_2}=\frac{p_1V_1}{2V_1}= \frac{p_1}{2}.\]

  In an isothermal process, all heat Q received by the gas is converted into the work W that it performs (internal energy U does not change). Thus, it holds true:

  \[Q=W.\]

• Numerical Substitution

  \[W=p_1V_1\,\ln\,2= 2\cdot{ 10^5}\cdot 1\cdot \ln\,2\,\mathrm{J}\dot{=} 138\,629\,\mathrm{J}\dot{=} 140\,\mathrm{kJ}\] \[p_2=\frac{p_1}{2}=\frac{2\cdot{ 10^5}}{2}\,\mathrm{Pa}= 10^5\,\mathrm{Pa}=100\,\mathrm{kPa}\] \[Q=W\dot{=}140\,\mathrm{kJ}\]

• Answer

  During the isothermal expansion, the air has performed work of approximately 140 kJ. Heat supplied to the gas is 140 kJ as well, and the resultant pressure of the gas is 100 kPa.