Question

A sample of chlorine gas is confined in a \(5.0-\mathrm{L}\) container at 328 torr and \(37^{\circ} \mathrm{C} .\) How many moles of gas are in the sample?

Short answer

0.0848 moles

Step-by-step solution

- Convert Temperature to Kelvin

Convert the temperature from Celsius to Kelvin using the formula: \[ T(K) = T(^{\text{C}}) + 273.15 \]Given: \[ T = 37^{\text{C}} \]So, \[ T(K) = 37 + 273.15 = 310.15 \text{ K} \]

- Convert Pressure to Atmospheres

Convert the given pressure from torr to atmospheres using the conversion factor: \[ 1 \text{ atm} = 760 \text{ torr} \]Given: \[ P = 328 \text{ torr} \]So, \[ P(\text{atm}) = \frac{328}{760} = 0.4316 \text{ atm} \]

- Use the Ideal Gas Law

Apply the Ideal Gas Law equation: \[ PV = nRT \]Where:\( P \) is the pressure in atmospheres (0.4316 atm)\( V \) is the volume in liters (5.0 L) \( n \) is the number of moles (unknown) \( R \) is the ideal gas constant (0.0821 L atm K⁻¹ mol⁻¹) \( T \) is the temperature in Kelvin (310.15 K)Rearrange the equation to solve for \( n \): \[ n = \frac{PV}{RT} \]

- Calculate the Number of Moles

Insert the known values into the equation: \[ n = \frac{(0.4316 \text{ atm})(5.0 \text{ L})}{(0.0821 \text{ L atm K⁻¹ mol⁻¹})(310.15 \text{ K})} \]Calculate: \[ n = \frac{2.158}{25.4511} = 0.0848 \text{ moles} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the relationship between the pressure, volume, temperature, and number of moles of a gas. The law is usually stated as: \[ PV = nRT \] where:
- \(P\) stands for pressure in atmospheres (atm).
- \(V\) represents volume in liters (L).
- \(n\) is the number of moles of the gas.
- \(R\) is the ideal gas constant, which is approximately 0.0821 L atm K⁻¹ mol⁻¹.
- \(T\) represents temperature in Kelvin (K).
The Ideal Gas Law allows us to predict one of these properties if we know the others. For instance, if we know the pressure, volume, and temperature of a gas, we can solve for the number of moles. This forms the basis for many calculations in chemistry.

Temperature Conversion

Temperature is often given in degrees Celsius (°C) in everyday situations. However, for scientific calculations, we use temperature in Kelvin (K). Converting Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature.
In our exercise, the temperature given is 37°C. To convert this to Kelvin:
\[T(K) = T(^{\text{C}}) + 273.15 = 37 + 273.15 = 310.15 \text{ K}\]
This is important because the Ideal Gas Law requires the temperature to be in Kelvin for accurate calculations.

Pressure Conversion

Pressure can be measured in various units, with torr and atmospheres (atm) being common ones in chemistry problems. Since the Ideal Gas Law uses pressure in atmospheres, converting from torr to atm is necessary if the given pressure is in torr. The conversion factor is:
\[ 1 \text{ atm} = 760 \text{ torr} \]
For the given pressure of 328 torr:
\[ P(\text{atm}) = \frac{328}{760} = 0.4316 \text{ atm}\]
This step ensures that the pressure is in the correct unit for using the Ideal Gas Law.

Number of Moles Calculation

To find the number of moles of a gas in a sample, we rearrange the Ideal Gas Law to solve for \( n \). The rearranged equation is:
\[ n = \frac{PV}{RT} \]
Using the values from the problem:
- \(P\) = 0.4316 atm
- \(V\) = 5.0 L
- \(R\) = 0.0821 L atm K⁻¹ mol⁻¹
- \(T\) = 310.15 K
We substitute these into the formula:
\[ n = \frac{(0.4316 \text{ atm})(5.0 \text{ L})}{(0.0821 \text{ L atm K⁻¹ mol⁻¹})(310.15 \text{ K})} = \frac{2.158}{25.4511} = 0.0848 \text{ moles}\]
This calculation shows how we use the Ideal Gas Law to determine the number of moles from the given pressure, volume, and temperature.