The critical pressure `P_(C)` and critical temperature `T_(C)` for a gas obeying van der Waal's equation are 80 atm at `87^(@)`C. molar mass of the gas is 130 g/mole. The compressibility factor for the above gas will be smaller than unity under the following condition:

Options

A. 1 atm and `800^(@)`C
B. 1 atm and `1200^(@)`C
C. 1 atm and `1000^(@)`C
D. 1 atm and `1100^(@)`C

Explanation

To determine the conditions under which the compressibility factor \( Z \) for the given gas is smaller than unity, we can follow these steps:

### Step 1: Understand the Critical Parameters
We are given:
- Critical pressure \( P_C = 80 \, \text{atm} \)
- Critical temperature \( T_C = 87^\circ C \)

### Step 2: Convert Critical Temperature to Kelvin
To convert the critical temperature from Celsius to Kelvin:
\[ T_C = 87 + 273 = 360 \, \text{K} \]

### Step 3: Identify the Compressibility Factor
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{RT} \]
For ideal gases, \( Z = 1 \). For real gases, \( Z \) can be greater than or less than 1 depending on the conditions.

### Step 4: Determine Boyle's Temperature
Boyle's temperature \( T_B \) is the temperature at which a real gas behaves like an ideal gas. It can be calculated using the relationship between critical temperature and Boyle's temperature:
\[ T_C = \frac{8}{27} \frac{a}{Rb} \]
Where \( a \) and \( b \) are the van der Waals constants. We can rearrange this to find \( T_B \):
\[ T_B = \frac{27}{8} T_C \]

### Step 5: Calculate Boyle's Temperature
Substituting \( T_C = 360 \, \text{K} \):
\[ T_B = \frac{27}{8} \times 360 = \frac{9720}{8} = 1215 \, \text{K} \]

### Step 6: Convert Boyle's Temperature to Celsius
To convert \( T_B \) back to Celsius:
\[ T_B = 1215 - 273 = 942 \, \text{°C} \]

### Step 7: Determine Conditions for \( Z < 1 \)
For the compressibility factor \( Z \) to be less than 1, the temperature must be less than Boyle's temperature:
\[ T < T_B \]
Thus, the condition is:
\[ T < 942 \, \text{°C} \]

### Step 8: Evaluate Given Options
We need to find a temperature below 942 °C from the options provided:
1. 1 atm and 800 °C
2. 1 atm and 1200 °C
3. 1 atm and 1000 °C
4. 1 atm and 1100 °C

The only option that satisfies \( T < 942 \, \text{°C} \) is:
- **1 atm and 800 °C**

### Final Answer
The compressibility factor \( Z \) for the gas will be smaller than unity when the temperature is **1 atm and 800 °C**.

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