Question

At what gas temperature trms would the root-mean-square (rms) speed of a helium atom be equal to that of an oxygen molecule in a gas at 300 k?

Answer 1

The rms speed of a gas is given by
$v_{rms} = \sqrt{ \frac{3RT}{M_m} }$ (1)
where
R is the gas constant
T is the absolute temperature of the gas
$M_m$ is the molar mass of the gas

For the oxygen molecules in the problem, T=300 K, and the molar mass of molecular oxygen is $M_m = 32 g/mol = 0.032 kg/mol$, therefore the rms speed of the molecules is
$v_{rms}= \sqrt{ \frac{3 (8.31 J/molK)(300 K)}{0.032 kg/mol} }=483.4 m/s$

We want the atoms of helium to have the same rms speed. So we can re-arrange equation (1), and using $v_{rms}=483.4 m/s$ and $M_m=4 g/mol=0.004 kg/mol$ (molar mass of helium) in order to find the helium atom temperature T:
$T= \frac{v_{rms}^2 M_m}{3 R}= \frac{(483.4 m/s)^2(0.004 kg/mol)}{3 (8.31 J/mol K)} =37.5 K$

Answer 2

Temperature of Helium atom is 37.5 K

$\texttt{ }$

Further explanation

Let's recall Work Done by Ideal Gas and Root-Mean-Square Speed of Gas formula as follows:

$\boxed{ W = \int {P} \, dV }$

where:

P = Gas Pressure ( Pa )

V = Gas Volume ( m³ )

$\texttt{ }$

$\boxed{ v_{rms} = \sqrt{ \frac{3RT}{M} }}$

where:

v_rms = root mean square speed ( m/s )

R = gas constant ( J/mol K )

T = temperature ( K )

M = molar mass of gas ( kg / mol )

Let us now tackle the problem!

$\texttt{ }$

Given:

molar mass of Helium = M_He = 4 gram/mol

molar mass of Oxygen = M_O₂ = 32 gram/mol

temperature of Oxygen = T_O₂ = 300 K

Asked:

temperature of Helium = T_He = ?

Solution:

$v_{He} = v_{O_2}$

$\sqrt{ \frac{3RT_{He}}{M_{He}} } = \sqrt{ \frac{3RT_{O_2}}{M_{O_2}} }$

$\sqrt{ \frac{T_{He}}{M_{He}} } = \sqrt{ \frac{T_{O_2}}{M_{O_2}}}$

$\sqrt{ \frac{T_{He}}{4 }} = \sqrt{ \frac{300}{32}}$

$\frac{T_{He}}{4 } = \frac{300}{32}}$

$T_{He} = \frac{4}{32} \times 300$

$\boxed{T_{He} = 37.5 \texttt{ K}}$

$\texttt{ }$

Learn more

• Buoyant Force : https://brainly.com/question/13922022

• Kinetic Energy : https://brainly.com/question/692781

• Volume of Gas : https://brainly.com/question/12893622

• Impulse : https://brainly.com/question/12855855
• Gravity : https://brainly.com/question/1724648

$\texttt{ }$

Answer details

Grade: High School

Subject: Physics

Chapter: Thermodynamics