COMEDK 2024 Shift 2 Paper

To find the ratio ‌

n1

n2

of moles of hydrogen to helium, we need to use the formula for the internal energy of an ideal gas. For a gas, internal energy (U) is given by:
U=‌

f

2

nRT
where:
f is the degrees of freedom
n is the number of moles
R is the universal gas constant
T is the temperature
For diatomic hydrogen (H2), the degrees of freedom at room temperature are 5(3 translational +2 rotational), so the internal energy is:
UH2=‌

5

2

n1RT
For monatomic helium (He), the degrees of freedom are 3 (all translational), so the internal energy is:
UHe=‌

3

2

n2R⋅2T=3n2RT
We are given that the internal energy of n1 moles of hydrogen at temperature T is equal to the internal energy of n2 moles of helium at temperature 2T. Therefore, we set the internal energies equal to each other:
‌

5

2

n1RT=3n2RT
We can cancel R and T from both sides of the equation:
‌

5

2

n1=3n2
Solving for the ratio ‌

n1

n2

, we multiply both sides by 2 :
5n1=6n2
Dividing both sides by 6 :
‌

n1

n2

=‌

6

5

Therefore, the correct answer is:
Option A: ‌

6

5