Question 20.1: A Tank of Helium A tank used for filling helium balloons has...
A Tank of Helium
A tank used for filling helium balloons has a volume of 0.300 m³ and contains 2.00 mol of helium gas at 20.0°C. Assume the helium behaves like an ideal gas.
(A) What is the total translational kinetic energy of the gas molecules?
(B) What is the average kinetic energy per molecule?
(A) Conceptualize Imagine a microscopic model of a gas in which you can watch the molecules move about the container more rapidly as the temperature increases. Because the gas is monatomic, the only type of motion the particles of the gas can exhibit is translation, and the total translational kinetic energy of the molecules is the internal energy of the gas.
Categorize We evaluate parameters with equations developed in the preceding discussion, so this example is a substitution problem.
Use Equation 20.21 with n = 2.00 mol and T = 293 K:
K_{\text {tot trans }}=N\left(\frac{1}{2} m_0 \overline{v^2}\right)=\frac{3}{2} N k_{B} T=\frac{3}{2} n R T (20.21)
\begin{aligned}E_{\text {int }} & =K_{\text {tot trans }}=\frac{3}{2} n R T=\frac{3}{2}(2.00 \text{ mol})(8.31 J/ \text{ mol} \cdot K)(293 K) \\& =7.30 \times 10^3 J\end{aligned}(B) Use Equation 20.19:
\frac{1}{2} m_0 \overline{v^2}=\frac{3}{2} k_{B} T (20.19)
\begin{aligned}K_{\text {avg }} & =\frac{1}{2} m_0 \overline{v^2}=\frac{3}{2} k_{B} T=\frac{3}{2}\left(1.38 \times 10^{-23} J/K\right)(293 K) \\& =6.07 \times 10^{-21} J\end{aligned}