Ideal gases & energy balances

Example problem set

These are sample problems with illustrative solutions. Replace or delete them when adding your own material. Use $R=8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$.

Foundations

Problem 1Easy

Calculate the pressure of $1.00\,\mathrm{mol}$ of ideal gas at $300\,\mathrm{K}$ in a volume of $0.0250\,\mathrm{m^3}$.

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Use the ideal gas equation:

$$P=\frac{nRT}{V}=\frac{1.00\times 8.314\times 300}{0.0250} =9.98\times10^4\,\mathrm{Pa}.$$

Problem 2Medium

A fixed amount of ideal gas is heated from $300\,\mathrm{K}$ to $450\,\mathrm{K}$ at constant pressure. Its initial volume is $2.00\,\mathrm{L}$. Find its final volume.

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At constant pressure, $V/T$ is constant:

$$V_2=V_1\frac{T_2}{T_1}=2.00\frac{450}{300}=3.00\,\mathrm{L}.$$

Energy and work

Problem 3Hard

A closed gas system expands from $2.00$ to $5.00\,\mathrm{L}$ against a constant external pressure of $100\,\mathrm{kPa}$ while absorbing $500\,\mathrm{J}$ of heat. Neglect kinetic and potential energy changes. Find the change in internal energy.

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Use work done on the gas as positive. Since $1\,\mathrm{kPa\,L}=1\,\mathrm{J}$:

$$w=-P_{\mathrm{ext}}\Delta V=-100(5.00-2.00)=-300\,\mathrm{J}, \qquad \Delta U=q+w=200\,\mathrm{J}.$$

Challenge

Problem 4Olympiad

One mole of monatomic ideal gas begins at $T_1=300\,\mathrm{K}$ and volume $V_1$. Compare the final temperatures after expansion to $2V_1$ by (a) reversible adiabatic expansion and (b) adiabatic free expansion into a vacuum in an insulated rigid vessel. Assume $C_V=3R/2$ and equilibrium endpoints.

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For reversible adiabatic expansion, $TV^{\gamma-1}$ is constant and $\gamma=5/3$:

$$T_{2,a}=300\left(\frac{1}{2}\right)^{2/3}\approx189\,\mathrm{K}.$$

For free expansion, $q=0$ and $w=0$, so $\Delta U=0$. Internal energy of an ideal gas depends only on temperature; therefore $T_{2,b}=300\,\mathrm{K}$.

The reversible adiabatic relation cannot be applied to the irreversible free expansion.