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 :

• A. $ \frac{3}{4} N_a k_B \left( \frac{T_2}{T_1} \right)$
• B. $ \frac{3}{8} N_a k_B \left( T_2 - T_1 \right)$
• C. $ \frac{3}{2} N_a k_B \left( T_2 - T_1 \right)$
• D. $ \frac{3}{4} N_a k_B \left( T_2 - T_1 \right)$

Answer 1

The Correct Option is B

To determine 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, we need to consider the properties and heat capacities of Helium.

Helium is a monoatomic ideal gas. For ideal gases, the heat capacity at constant volume (C_V) is given by:

C_V = \frac{3}{2} R

Where R is the universal gas constant.

Since we are dealing with 1 gram of helium, we need the specific heat capacity at constant volume. Helium's molar mass is approximately 4 g/mol, so for 1 gram of helium:

C' = \frac{C_V}{\text{molar mass of Helium}} = \frac{3}{2} \frac{R}{4}

The heat energy required (Q) when the temperature changes from T_1 to T_2 can be calculated using:

Q = m \cdot C' \cdot (T_2 - T_1)

Where m is the mass. For our specific case, m = 1 g.

Q = \frac{3}{2} \cdot \frac{R}{4} \cdot (T_2 - T_1)

Replacing R with N_a \cdot k_B (where N_a is Avogadro's number and k_B is the Boltzmann constant) gives us:

Q = \frac{3}{2} \cdot \frac{N_a \cdot k_B}{4} \cdot (T_2 - T_1)

Simplifying this expression results in the heat energy formula:

Q = \frac{3}{8} N_a k_B (T_2 - T_1)

Therefore, the correct answer is \frac{3}{8} N_a k_B (T_2 - T_1). This aligns with option 2 provided in the question.