Question

The two containers of gas in FIGURE Q20.8 are in good thermal contact with each other but well insulated from the environment. They have been in contact for a long time and are in thermal equilibrium.

a. Is $v_{\text{rms}}$of helium greater than, less than, or equal to $v_{\text{rms}}$of argon? Explain.

b. Does the helium have more thermal energy, less thermal energy, or the same amount of thermal energy as the argon? Explain.

Short answer

a. The $v_{\text{rms}}$of helium is greater than that of argon.

b. In comparison to argon, helium has a lower thermal energy.

Step-by-step solution

Concept Introduction (Part a)

An electromotive force$\left(v_{rms}\right)$ is characterized by the root-mean-square voltage $\left(rms\right)$ of its sinusoidal output. It is the square root of the voltage squared averaged over time.

Explanation (Part a)

The two gases are in thermal equilibrium which implies they have the same temperature $T.$

The expression for root means square speed in terms of temperature is

$v_{\mathrm{rms}}=\sqrt{\frac{3k_{\mathrm{B}}T}{m\text{molecule}}}.$

Therefore

$v_{\text{rms}}\propto \sqrt{T},v_{\text{rms}}\propto \frac{1}{\sqrt{m_{\text{molecule}}}}$

Final Answer (Part a)

Since $v_{rms}$and the square root of mass is inversely proportional, helium's RMS speed is greater than argon's. In any case, the two gasses have the same temperature, but argon's atoms are heavier.

Concept Introduction (Part b)

Combined kinetic energies in a gas correspond to the total thermal energy. However, thermal energy is not the same as heat. Heat is a way to move energy from one place to another.

Explanation (Part a)

Considering ideal gases, the following expression describes their thermal energy:

$E_{\text{th}}=n{C}_{V}T,$

The molar specific heat at constant volume is therefore proportional to temperature, molecular number, and molecular weight.

Since those parameters remain the same for both gases, we don't need to state $T$.

Final Answer (Part b)

Helium and argon gases belong to the same class of monoatomic molecules, so they are also characterized by the same $C_V.$ Helium and Argon both have the same number of moles, but Helium has lower thermal energy.