One mole of a certain ideal gas obtains an amount of heat `Q=1.60kJ` when its temperature is increased by `DeltaT=72K`, keeping its pressure constant. The vlaue of `(C_(P))/(C_(V))` for the gas is

Options

A. (a)`1.60`
B. (b)`1.40`
C. (c)`1.50`
D. `1.30`

Answer

To solve the problem, we need to find the value of \(\frac{C_P}{C_V}\) for the ideal gas given the amount of heat \(Q\) and the change in temperature \(\Delta T\) at constant pressure.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amount of heat \(Q = 1.60 \, \text{kJ} = 1600 \, \text{J}\) (since \(1 \, \text{kJ} = 1000 \, \text{J}\))
- Change in temperature \(\Delta T = 72 \, \text{K}\)
- Number of moles \(n = 1 \, \text{mol}\)

2. **Use the Formula for Heat at Constant Pressure:**
The heat absorbed by the gas at constant pressure is given by:
\[ Q = n C_P \Delta T \]
Rearranging this formula to solve for \(C_P\):
\[ C_P = \frac{Q}{n \Delta T} \]

3. **Substitute the Known Values:**
\[ C_P = \frac{1600 \, \text{J}}{1 \, \text{mol} \times 72 \, \text{K}} = \frac{1600}{72} \]

4. **Calculate \(C_P\):**
\[ C_P = \frac{1600}{72} \approx 22.22 \, \text{J/mol K} \]

5. **Use the Relation Between \(C_P\) and \(C_V\):**
The relation between the molar specific heats is given by:
\[ C_P - C_V = R \]
where \(R\) is the gas constant, \(R \approx 8.31 \, \text{J/mol K}\).

6. **Rearranging to Find \(C_V\):**
\[ C_V = C_P - R = 22.22 \, \text{J/mol K} - 8.31 \, \text{J/mol K} \]
\[ C_V \approx 13.91 \, \text{J/mol K} \]

7. **Calculate the Ratio \(\frac{C_P}{C_V}\):**
\[ \frac{C_P}{C_V} = \frac{22.22}{13.91} \approx 1.60 \]

### Final Answer:
The value of \(\frac{C_P}{C_V}\) for the gas is approximately **1.60**.