Question

Molar volume of \(\mathrm{CO}_{2}\) is maximum at (1) STP (2) \(0^{\circ} \mathrm{C}, 2 \mathrm{~atm}\) (3) \(127^{\circ} \mathrm{C}, \mathrm{latm}\) (4) \(273^{\mathrm{p}} \mathrm{C}, 2\) atm

Short answer

127°C, 1 atm

Step-by-step solution

Understanding Molar Volume

Molar volume refers to the volume occupied by one mole of a gas under specified conditions of temperature and pressure. It can be calculated using the Ideal Gas Law.

Use the Ideal Gas Law

The Ideal Gas Law is given by \[ PV = nRT \]. Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

Express Molar Volume

For one mole of gas (\( n = 1 \)), the equation becomes \[ V = \frac{RT}{P} \]

Compare Conditions

To identify the condition where molar volume is maximum, look at the expression \( V = \frac{RT}{P} \). The molar volume will be maximum when \( T \) is highest and \( P \) is lowest. Let's compare each given condition: (1) STP: \(0^{\text{o}} \text{C}\text{, 1 atm}\) (2) \(0^{\text{o}} \text{C}\text{, 2 atm}\) (3) \(127^{\text{o}} \text{C}\text{, 1 atm}\) (4) \(273^{\text{o}} \text{C}\text{, 2 atm}\)

Calculate Relevant Ratios

By substituting the values in the equation \( V = \frac{RT}{P} \), we can observe: (1) STP: \( V = \frac{R \times 273}{1} \) (2) \(0^{\text{o}} \text{C}\text{, 2 atm}\): \( V = \frac{R \times 273}{2} \) (3) \(127^{\text{o}} \text{C}\text{, 1 atm}\): \( V = \frac{R \times 400}{1} \) (4) \(273^{\text{o}} \text{C}\text{, 2 atm}\): \( V = \frac{R \times 546}{2} \)

Conclusion

Among these, the expression in condition (3) \(127^{\text{o}} \text{C}\text{, 1 atm}\) results in the highest value. So, the molar volume of \(\text{CO}_2\) is maximum at \(127^{\text{o}} \text{C}\text{, 1 atm}\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the behavior of ideal gases. It is given by the formula: \[ PV = nRT \]. Each symbol in the equation has a specific meaning: \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) is the gas constant, and \( T \) is the temperature. This equation connects four different properties of a gas, allowing us to predict one property if we know the other three. For molar volume calculations, the Ideal Gas Law is particularly useful. When considering one mole of gas ( = 1), you can simplify the equation to: \[ V = \frac{RT}{P} \]. This formula helps us understand how volume changes with temperature and pressure. Remember, the Ideal Gas Law assumes that gas molecules do not interact and occupy no volume, which is a good approximation under many conditions, especially at high temperature and low pressure.

Temperature and Pressure Conditions

In the context of gases, temperature and pressure are crucial factors that influence volume. Standard Temperature and Pressure (STP) is a common reference point in chemistry, defined as \(0^{\text{o}}C\) (273 K) and 1 atm pressure. When analyzing the effect of temperature and pressure on gas volume, it's essential to remember how they affect the Ideal Gas Law equation. At higher temperatures, gas molecules have more kinetic energy, which can lead to increased volume when pressure is constant. Conversely, higher pressure results in a decrease in volume for a constant temperature. These inverse relationships are key to understanding why the molar volume of a gas is maximized at high temperatures and low pressures. For example, if we compare:

• STP: 273 K, 1 atm
• 273 K, 2 atm
• 400 K, 1 atm
• 546 K, 2 atm

The molar volume will be the greatest when temperature is at its highest (546 K) and pressure is the lowest (1 atm for condition 3).

Molar Volume Calculations

Molar volume is the volume occupied by one mole of a gas. To calculate it, we utilize the simplified Ideal Gas Law: \[ V = \frac{RT}{P} \]. When you break down this equation for different conditions, it becomes clear how temperature and pressure influence molar volume. Let's examine the given scenarios:(1) **STP (0°C, 1 atm)**:
\[ V = \frac{R \times 273}{1} \](2) **0°C, 2 atm**:
\[ V = \frac{R \times 273}{2} \](3) **127°C, 1 atm**:
\[ V = \frac{R \times 400}{1} \](4) **273°C, 2 atm**:
\[ V = \frac{R \times 546}{2} \]By comparing these calculated volumes, it's evident that scenario (3) provides the maximum volume. At 127°C (400 K) and 1 atm pressure, the molar volume achieves its highest value. This example highlights the importance of considering both temperature and pressure when calculating molar volumes. Exploring these conditions allows for a deeper understanding of gas behavior and reinforces the principle that gas volume increases with higher temperature and decreases with higher pressure.