Question

At what temperature does the average translational kinetic energy of a molecule in a gas becomes equal to kinetic energy of an electron accelerated from rest through potential difference of $V$ volt? [$N$ = number of molecules, $R$ = gas constant, $e$ = electronic charge]

• A. $\frac{2eVN}{3R}$
• B. $\frac{eVN}{R}$
• C. $\frac{eVN}{4R}$
• D. $\frac{3eVN}{2R}$

Hint:

To easily remember thermal energy values, associate the fraction $\frac{3}{2}$ with a molecule's 3 linear degrees of freedom ($\frac{1}{2}k_BT$ per degree). Moving the $\frac{3}{2}$ fraction to the other side of an equation always flips it to $\frac{2}{3}$, instantly pointing you to an answer that must contain a 3 in the denominator.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem requires us to calculate the absolute temperature ($T$) at which the average translational kinetic energy of a gas molecule balances the kinetic energy gained by a single electron accelerated across an electrical potential difference of $V$ volts.

Step 2: Key Formula or Approach:
1. The kinetic energy acquired by an electron with charge $e$ when accelerated from rest across a potential difference $V$ is given by:
$$E_e = eV$$ 2. The average translational kinetic energy of a single gas molecule at absolute temperature $T$ is given by the kinetic theory of gases:
$$E_g = \frac{3}{2} k_B T$$ where $k_B$ is the Boltzmann constant.
3. The relationship between the universal gas constant $R$, Avogadro's number or total molecular reference count $N$, and $k_B$ is:
$$k_B = \frac{R}{N}$$

Step 3: Detailed Explanation:
Equate the two energy expressions as specified by the problem statement:
$$\frac{3}{2} k_B T = eV$$ Substitute the relationship $k_B = \frac{R}{N}$ into this balance equation:
$$\frac{3}{2} \left(\frac{R}{N}\right) T = eV$$ Rearrange terms to isolate the temperature parameter $T$ on the left side:
$$T = \frac{2 \cdot e \cdot V \cdot N}{3R} = \frac{2eVN}{3R}$$

Step 4: Final Answer:
The required temperature value is $\frac{2eVN}{3R}$, which corresponds to option (A).