175 calories of heat is reuired to raise the temperature of 5 mol of an ideal gas at constant pressure from `20^@ C ` to `25 ^@ C ` the amount of heat required to raise the temperature of same gas from `20^@ C ` to `25^@ C ` at constant volume will be

Options

A. 100 cal
B. 125 cal
C. 150 cal
D. 175 cal

Answer

To solve the problem of finding the amount of heat required to raise the temperature of 5 moles of an ideal gas from 20°C to 25°C at constant volume, we can follow these steps:

### Step 1: Understand the Given Data
We know the following:
- Heat required at constant pressure (Qp) = 175 calories
- Number of moles (n) = 5 moles
- Temperature change (ΔT) = 25°C - 20°C = 5°C

### Step 2: Write the Heat Exchange Equations
For heat exchange at constant pressure:
\[ Q_p = n C_p \Delta T \]
For heat exchange at constant volume:
\[ Q_v = n C_v \Delta T \]

### Step 3: Relate Qp and Qv
We can relate the two equations as follows:
\[ \frac{Q_v}{Q_p} = \frac{C_v}{C_p} \]

### Step 4: Calculate Cp from Qp
From the equation for Qp, we can express Cp:
\[ C_p = \frac{Q_p}{n \Delta T} \]
Substituting the known values:
\[ C_p = \frac{175 \text{ calories}}{5 \text{ moles} \times 5°C} = \frac{175}{25} = 7 \text{ calories/(mole°C)} \]

### Step 5: Use the Relation Cp - Cv = R
We know that:
\[ C_p - C_v = R \]
Where R is the gas constant. For ideal gases, we can use the value of R in calories:
\[ R = 2 \text{ calories/(mole°C)} \]
Thus:
\[ C_v = C_p - R = 7 \text{ calories/(mole°C)} - 2 \text{ calories/(mole°C)} = 5 \text{ calories/(mole°C)} \]

### Step 6: Calculate Qv
Now we can find Qv using the equation for heat exchange at constant volume:
\[ Q_v = n C_v \Delta T \]
Substituting the values:
\[ Q_v = 5 \text{ moles} \times 5 \text{ calories/(mole°C)} \times 5°C = 125 \text{ calories} \]

### Final Answer
The amount of heat required to raise the temperature of the gas from 20°C to 25°C at constant volume is **125 calories**.

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