A given amount of gas occupies 10.0dm$^3$ at 4 atm and 273°C. The number of moles of the gas present is? [Molar volume of gas at s.t.p = 22.4 dm$^3$]

Analysis

To solve this problem, we need to determine the number of moles ($n$) of a gas given its volume ($V$), pressure ($P$), and temperature ($T$). We will use the Ideal Gas Law equation:

Concept

The Ideal Gas Law relates the pressure, volume, temperature, and the number of moles of a gas.

• Pressure ($P$): 4 atm
• Volume ($V$): 10.0 dm³
• Temperature ($T$): Must be in Kelvin. $T(K) = 273 + 273 = 546 \text{ K}$
• Gas Constant ($R$): Since pressure is in atm and volume is in dm³, we use $R = 0.0821 \text{ dm}^3\text{atm K}^{-1}\text{mol}^{-1}$

Alternatively, we can use the relationship between the given state and Standard Temperature and Pressure (STP) using the Combined Gas Law, but the Ideal Gas Law is more direct here.

Solution

Step 1: Convert temperature to Kelvin

Step 2: Apply the Ideal Gas Law formula

Step 3: Substitute the values

Alternative Method (Using STP): At STP ($P = 1 \text{ atm}, T = 273 \text{ K}$), 1 mole occupies 22.4 dm³. Using the gas law $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$:

Since 1 mole = 22.4 dm³, then $n = \frac{20}{22.4} \approx 0.893 \text{ mol}$.

Both methods confirm the result is approximately 0.89 mol.

Conclusion

The calculated number of moles is 0.89 mol, which matches option A.

Exam Tip

Always remember to convert Celsius to Kelvin by adding 273 before plugging your temperature into any gas law equation. Also, check your units—if the volume is in dm³ and pressure in atm, ensure your gas constant ($R$) matches those units!