The amount of heat supplied to `4xx10^(-2)` kg of nitrogen at room temperature to rise its temperature by `50^(@)C` at constant pressure is (Molecular mass of nitrogen is 28 and `R= 8.3 J mol^(-1)K^(-1))`

Options

A. `2.08 KJ`
B. `3.08 KJ`
C. `4.08 KJ`
D. `5.08 KJ`

Explanation

To solve the problem of calculating the amount of heat supplied to `4 x 10^(-2)` kg of nitrogen at constant pressure to raise its temperature by `50°C`, we can follow these steps:

### Step 1: Convert the mass of nitrogen to grams
Given mass of nitrogen:
\[ m = 4 \times 10^{-2} \text{ kg} = 40 \text{ g} \]

### Step 2: Calculate the number of moles of nitrogen
Using the molecular mass of nitrogen (N₂) which is 28 g/mol, we can calculate the number of moles (N) using the formula:
\[ N = \frac{\text{mass}}{\text{molar mass}} = \frac{40 \text{ g}}{28 \text{ g/mol}} = \frac{10}{7} \text{ mol} \]

### Step 3: Identify the change in temperature
The change in temperature (ΔT) is given as:
\[ \Delta T = 50°C \]

### Step 4: Calculate the specific heat capacity at constant pressure (Cp)
For a diatomic gas like nitrogen, the specific heat capacity at constant volume (Cv) is given by:
\[ C_v = \frac{5R}{2} \]
Thus, the specific heat capacity at constant pressure (Cp) is:
\[ C_p = C_v + R = \frac{5R}{2} + R = \frac{7R}{2} \]

### Step 5: Calculate the heat supplied (Q)
The formula for heat supplied at constant pressure is:
\[ Q = N C_p \Delta T \]
Substituting the values we have:
\[ Q = \left(\frac{10}{7} \text{ mol}\right) \left(\frac{7R}{2}\right) (50 \text{ K}) \]
The 7 in the numerator and denominator cancels out:
\[ Q = \left(\frac{10}{2}\right) R (50) = 5R \times 50 = 250R \]

### Step 6: Substitute the value of R
Given \( R = 8.3 \, \text{J/mol·K} \):
\[ Q = 250 \times 8.3 \, \text{J} = 2075 \, \text{J} \]

### Step 7: Convert the heat to kilojoules
To convert joules to kilojoules:
\[ Q = \frac{2075 \, \text{J}}{1000} = 2.075 \, \text{kJ} \]

### Final Answer
Thus, the amount of heat supplied is approximately:
\[ Q \approx 2.08 \, \text{kJ} \]

---