In the equation of state of an ideal gas PV =nRT , the value of the universal gas constant is not correct :

Options

A. `"8.314 JK"^(-1)"mol"^(-1)`
B. `0.0821" atm L mol"^(-1)K^(-1)`
C. `0.8314" bar L mol"^(-1)K^(-1)`
D. `"2 cal mol"^(-1)K^(-1)`

Answer

To solve the problem regarding the incorrect value of the universal gas constant \( R \) in the equation of state of an ideal gas \( PV = nRT \), we will follow these steps:

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature of the gas in Kelvin

### Step 2: Identify Standard Conditions
Standard Temperature and Pressure (STP) are defined as:
- Standard Temperature = 273 K
- Standard Pressure = 1 atm = 101.325 kPa

At STP, the volume of 1 mole of an ideal gas is approximately:
\[ V = 22.4 \text{ L} \]

### Step 3: Calculate the Value of \( R \)
Using the values at STP, we can calculate \( R \):
1. Substitute the values into the ideal gas law:
\[ P = 1 \text{ atm} = 101.325 \text{ kPa} \]
\[ V = 22.4 \text{ L} = 22.4 \times 10^{-3} \text{ m}^3 \]
\[ n = 1 \text{ mole} \]
\[ T = 273 \text{ K} \]

2. Rearranging the ideal gas law for \( R \):
\[ R = \frac{PV}{nT} \]

3. Substitute the values:
\[ R = \frac{(101.325 \times 10^3 \text{ Pa})(22.4 \times 10^{-3} \text{ m}^3)}{(1 \text{ mole})(273 \text{ K})} \]

4. Calculate \( R \):
\[ R \approx \frac{2265.12 \text{ J}}{273} \approx 8.314 \text{ J/(mol K)} \]

### Step 4: Convert \( R \) to Other Units
1. **Convert to atm·L/(mol·K)**:
\[ R = 0.0821 \text{ atm·L/(mol·K)} \]

2. **Convert to calorie**:
\[ 1 \text{ calorie} = 4.184 \text{ J} \]
\[ R \approx \frac{8.314 \text{ J/(mol K)}}{4.184 \text{ J/cal}} \approx 1.988 \text{ cal/(mol K)} \]

### Step 5: Identify the Incorrect Value
Now we compare the calculated values of \( R \) with the given options. The correct values are:
- \( R \approx 0.0821 \text{ atm·L/(mol·K)} \)
- \( R \approx 8.314 \text{ J/(mol·K)} \)
- \( R \approx 1.988 \text{ cal/(mol·K)} \)

By checking the options provided in the question, we identify which one does not match the calculated values.

### Conclusion
The incorrect value of the universal gas constant \( R \) among the options provided is the one that deviates from the calculated values.

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