Question 901: Thermodynamics & KTG

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?
($$\mathrm{N}=$$ number of molecules, $$\mathrm{R}=$$ gas constant, $$\mathrm{c}=$$ electronic charge)

Accepted Answer

To determine the temperature at which the average translational kinetic energy of a molecule in a gas becomes equal to the kinetic energy of an electron accelerated from rest through a potential difference of 'V' volts, we start by comparing their respective kinetic energies.
The kinetic energy of an electron accelerated through a potential difference V is given by:
$$ E_{	ext{electron}} = eV $$
where:
$$e$$ is the electronic charge.
$$V$$ is the potential difference in volts.
For a molecule in an ideal gas, the average translational kinetic energy is given by:
$$ E_{	ext{translational}} = \frac{3}{2} k_B T $$
where:
$$k_B$$ is the Boltzmann constant.
$$T$$ is the absolute temperature in Kelvin.
Now, we equate the kinetic energy of the electron to the average translational kinetic energy of a molecule in the gas:
$$ eV = \frac{3}{2} k_B T $$
Solving for temperature T, we get:
$$ T = \frac{2 eV}{3 k_B} $$
For a gas with number of molecules $$N$$ and using the relationship between Boltzmann constant $$k_B$$ and gas constant $$R$$ (i.e., $$k_B = \frac{R}{N}$$), we substitute this value into the equation:
$$ T = \frac{2 eV}{3} \cdot \frac{N}{R} $$
Simplifying, we obtain:
$$ T = \frac{2 eVN}{3R} $$
Thus, the correct option is:
Option A
$$\frac{2 \mathrm{eVN}}{3 \mathrm{R}}$$

Answer

$$\frac{2 \mathrm{eVN}}{3 \mathrm{R}}$$

Answer

$$\mathrm{\frac{e V N}{R}}$$

Answer

$$\mathrm{\frac{e V N}{4 R}}$$

Answer

$$\mathrm{\frac{3 e V N}{2 R}}$$

Thermodynamics & KTG

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? ($\mathrm{N}= number of molecules, \mathrm{R}= gas constant, \mathrm{c}=$ electronic charge)

Choose the correct answer: