Question

An experiment calls for \(3.70 \mathrm{~mol}\) of chlorine, \(\mathrm{Cl}_{2}\) What volume will this be if the gas volume is measured at \(36^{\circ} \mathrm{C}\) and \(3.30 \mathrm{~atm} ?\)

Short answer

The volume is approximately 28.2 L.

Step-by-step solution

Understand the Ideal Gas Law

The Ideal Gas Law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin. We will use this formula to find the volume of chlorine gas.

Convert Temperature to Kelvin

The given temperature is \(36^{\circ} \mathrm{C}\). To convert Celsius to Kelvin, use the formula: \( T(K) = T(^{\circ}C) + 273.15 \). Therefore, \( 36 + 273.15 = 309.15 \mathrm{~K} \).

Use the Ideal Gas Constant

For pressure in atm and volume in liters, the ideal gas constant \( R \) is \( 0.0821 \mathrm{~L}\cdot\mathrm{atm}/\mathrm{mol}\cdot\mathrm{K} \). We will use this value in our calculation.

Solve for Volume \( V \) Using the Ideal Gas Law

Substitute the known values into the ideal gas law: \( n = 3.70 \mathrm{~mol} \), \( T = 309.15 \mathrm{~K} \), \( P = 3.30 \mathrm{~atm} \), \( R = 0.0821 \mathrm{~L}\cdot\mathrm{atm}/\mathrm{mol}\cdot\mathrm{K} \). The equation becomes: \[ V = \frac{nRT}{P} = \frac{3.70 \times 0.0821 \times 309.15}{3.30} \].

Key concepts

Molar Volume

Molar volume is a fundamental concept when dealing with gases. It refers to the volume that one mole of a gas occupies at a given temperature and pressure. Understanding molar volume is crucial, especially under standard conditions, which are defined as a temperature of 0°C (273.15 K) and a pressure of 1 atm.

• The molar volume of an ideal gas at these conditions is approximately 22.4 L/mol.
• However, when the conditions differ, such as in the case of the chlorine gas in our problem, the volume that a mole occupies will change accordingly.

The concept of molar volume is intricately linked with the Ideal Gas Law. It provides a convenient way to predict how much space a specific amount of substance will need based on the ideal behavior of gases. In our example, calculating the volume of 3.70 moles of chlorine requires adjusting for the given temperature and pressure conditions through the Ideal Gas Law.

Gas Constant

The gas constant, denoted as \( R \), is a critical value in the Ideal Gas Law equation. It serves as a bridge, connecting pressure, volume, temperature, and number of moles. For the Ideal Gas Law,

• When pressure is measured in atm, volume in liters, temperature in Kelvin, and amount in moles, the value of \( R \) is 0.0821 L·atm/(mol·K).
• This value is derived experimentally and is used universally for calculations involving ideal gases.

In our problem, using the value of \( R \) as 0.0821 ensures the units are consistent throughout the calculation. It's important to remember that \( R \) can have different numerical values depending on the units used for pressure and volume, but in this context, 0.0821 is the correct choice.

Temperature Conversion

Temperature conversion is a necessary step when using the Ideal Gas Law, as the formula requires temperature in Kelvin. Given the experiment's conditions, converting from Celsius to Kelvin is simple and essential.

• The conversion formula is: \( T(K) = T(°C) + 273.15 \).
• In our specific example with chlorine gas, the temperature of 36°C is converted to 309.15 K.

Using Kelvin ensures that the absolute temperature is measured, as temperature scales like Celsius are not suitable for thermodynamic calculations involving gases. This conversion is straightforward yet critical for ensuring accurate results in calculations.