To find the ratio
‌ 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=‌nRTwhere:
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=‌n1RT For monatomic helium (He), the degrees of freedom are 3 (all translational), so the internal energy is:
UHe=‌n2R⋅2T=3n2RTWe 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:
‌n1RT=3n2RT We can cancel
R and
T from both sides of the equation:
‌n1=3n2Solving for the ratio
‌, we multiply both sides by 2 :
5n1=6n2Dividing both sides by 6 :
‌=‌Therefore, the correct answer is:
Option A:
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