Question

The internal energy of 3 moles of hydrogen at temperature T Is equal to the internal energy of n moles of helium at temperature ${T}/{2}\;$. The value of n is (assume hydrogen and helium to behave like ideal gases).
$A.\,5$
$B.\,10$
$C.\,{3}/{2}\;$
$D.\,6$

Answer 1

The problem can be solved using the formula of the internal energy of an ideal gas. We need to separately compute the internal energies of the hydrogen and helium, and then we need to equate them as per the given condition. Finally, substitute the given values to obtain the value of n.

Formula used:

& {{E}_{{{H}_{2}}}}={}^{5}/{}_{2}nRT \\\ & {{E}_{He}}={}^{3}/{}_{2}nRT \\\ \end{aligned}$$ **Complete step-by-step answer:** Let us begin the calculation by finding the internal energy of the hydrogen and helium gas considering them as the ideal gas, one by one. So, we have, Now consider only the hydrogen gas. From the data, we have that data as follows. The number of moles of the hydrogen, n = 3 moles The temperature = T Let us compute the internal energy of the hydrogen gas. Being diatomic, the hydrogen gas has the internal energy formula as given below, $${{E}_{{{H}_{2}}}}={}^{5}/{}_{2}nRT$$ Substitute the given values in the above equation. $${{E}_{{{H}_{2}}}}={}^{5}/{}_{2}\times 3RT$$…… (1) Now consider only the helium gas. From the data, we have that data as follows. The number of moles of the hydrogen, n = n moles The temperature = $${}^{T}/{}_{2}$$ Let us compute the internal energy of the helium gas. Being monoatomic, the helium gas has the internal energy formula as given below, $${{E}_{He}}={}^{3}/{}_{2}nRT$$ Substitute the given values in the above equation. $${{E}_{He}}={}^{3}/{}_{2}\times nR\times {}^{T}/{}_{2}$$ …… (2) As per the given condition, we have to equate the internal energy of the helium gas and internal energy of the helium gas. So, we have to equate the equations (1) and (2), $${}^{5}/{}_{2}\times 3R\times T={}^{3}/{}_{2}\times nR\times {}^{T}/{}_{2}$$ Upon further continuing the calculation, we get, $$\begin{aligned} & 15T=3n\times {}^{T}/{}_{2} \\\ & \Rightarrow n=10 \\\ \end{aligned}$$ Therefore, the value of the number of moles of the helium gas is 10. **So, the correct answer is “Option B”.** **Note:** The things to be on your finger-tips for further information on solving these types of problems are: There are a total of three different formulae to compute the internal energy of the gases. They are based on the nature (presence of the atoms) of the gases, that is, monoatomic, diatomic and polyatomic.