The amount of heat energy required to raise the temperature of 1 g of Helium at NTP, from $T_1$ K to $T_2$ K i

Question

The amount of heat energy required to raise the temperature of 1 g of Helium at NTP, from $T_1$ K to $T_2$ K is:

Accepted Answer

1.  **Identify the Gas and Moles:** Helium (He) is a monoatomic gas. Its molar mass ($M$) is $4 	ext{ g/mol}$. Given mass $m = 1 	ext{ g}$.
    Number of moles $n = rac{m}{M} = rac{1}{4} 	ext{ mol}$.
2.  **Identify Specific Heat:** For a monoatomic gas, the molar specific heat at constant volume is $C_v = rac{3}{2}R$. (Note: While the problem mentions NTP, which often implies constant pressure, the available options match the calculation for constant volume or internal energy change. If calculated at constant pressure, $C_p = rac{5}{2}R$, yielding a coefficient of 5/8, which is not an option).
3.  **Calculate Heat Energy:** The heat energy required ($Q$) is given by $Q = n C_v \Delta T$.
    $Q = \left(rac{1}{4}
ight) 	imes \left(rac{3}{2}R
ight) 	imes (T_2 - T_1) = rac{3}{8} R (T_2 - T_1)$.
4.  **Substitute Constants:** The universal gas constant $R$ is related to the Boltzmann constant ($k_B$) and Avogadro's number ($N_a$) by the equation $R = N_a k_B$.
    Substituting this into the heat equation: $Q = rac{3}{8} N_a k_B (T_2 - T_1)$.
5.  **Correction Note:** The input text '111 g' is interpreted as a typo for '1 g' based on the mathematical derivation yielding the correct option.

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