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 :

Options

A. `(3)/(8) N_(a)K_(B) (T_(2) - T_(1))`
B. `(3)/(2) N_(a) K_(B) (T_(2) - T_(1))`
C. `(3)/(4) N_(a) K_(B) (T_(2) - T_(1))`
D. `(3)/(4) N_(a)K_(B) ((T_(2))/(T_(1)))`

Answer

To find 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 can follow these steps:

### Step 1: Identify the Formula
Since the volume of the gas remains constant, we use the formula for heat energy at constant volume:
\[ \Delta Q = n C_v \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_v \) = molar heat capacity at constant volume
- \( \Delta T = T_2 - T_1 \) = change in temperature

### Step 2: Determine the Number of Moles
Given that the mass of Helium is 1 g and the molar mass of Helium is approximately 4 g/mol, we can calculate the number of moles:
\[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{1 \text{ g}}{4 \text{ g/mol}} = \frac{1}{4} \text{ mol} \]

### Step 3: Find the Value of \( C_v \)
Helium is a monoatomic gas, and the molar heat capacity at constant volume \( C_v \) for a monoatomic ideal gas is given by:
\[ C_v = \frac{3}{2} R \]
where \( R \) is the universal gas constant.

### Step 4: Calculate \( \Delta T \)
The change in temperature is given by:
\[ \Delta T = T_2 - T_1 \]

### Step 5: Substitute Values into the Formula
Now, substituting \( n \), \( C_v \), and \( \Delta T \) into the heat energy formula:
\[ \Delta Q = n C_v \Delta T = \left(\frac{1}{4}\right) \left(\frac{3}{2} R\right) (T_2 - T_1) \]

### Step 6: Simplify the Expression
Now, simplify the expression:
\[ \Delta Q = \frac{3}{8} R (T_2 - T_1) \]

### Step 7: Final Expression
Thus, the amount of heat energy required to raise the temperature of 1 g of Helium from \( T_1 \) K to \( T_2 \) K is:
\[ \Delta Q = \frac{3}{8} R (T_2 - T_1) \]