Question

A \(0.1-\mathrm{m}^{3}\) cylinder contains a gaseous mixture with a molar composition of \(97 \% \mathrm{CO}\) and \(3 \% \mathrm{CO}_{2}\) initially at 138 bar. Due to a leak, the pressure of the mixture drops to 129 bar while the temperature remains constant at \(30^{\circ} \mathrm{C}\). Using Kay's rule, estimate the amount of mixture, in \(\mathrm{kmol}\), that leaks from the cylinder.

Short answer

0.30 kmol

Step-by-step solution

Calculate the initial number of moles

Start by using the ideal gas law to calculate the initial amount of moles in the cylinder. The ideal gas law is given by: \[ PV = nRT \] Where: - \(P\) is the pressure (in bar) - \(V\) is the volume (in m^3) - \(n\) is the number of moles - \(R\) is the gas constant (0.08314 bar·L·K⁻¹·mol⁻¹) - \(T\) is the temperature (in K). Convert the temperature to Kelvin: \[ T = 30 + 273.15 = 303.15 \, K \] Now, substitute the known values into the ideal gas law for the initial state: \[ 138 \times 0.1 = n \times 0.08314 \times 303.15 \] Solving for \(n\): \[ n = \frac{138 \times 0.1}{0.08314 \times 303.15} \approx 5.47 \text{ kmol} \]

Calculate the final number of moles

Using the same process, but now with the final pressure of 129 bar: Substituting the known values into the ideal gas law for the final state: \[ 129 \times 0.1 = n \times 0.08314 \times 303.15 \] Solving for \(n\): \[ n = \frac{129 \times 0.1}{0.08314 \times 303.15} \approx 5.17 \text{ kmol} \]

Calculate the amount of gas that leaked

The difference between the initial and final number of moles represents the amount of gas that leaked: \[ n_{leaked} = n_{initial} - n_{final} \] Substituting the values: \[ n_{leaked} = 5.47 - 5.17 = 0.30 \text{ kmol} \]

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics. It provides a relationship between pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas. The equation is expressed as: \[ PV = nRT \]. Here, R is the universal gas constant, valued at 0.08314 bar·L·K⁻¹·mol⁻¹. Understanding this equation is crucial because it allows us to calculate one of the variables if the others are known. For example, in the given exercise, we use the ideal gas law to find the initial and final moles of gas in the cylinder. This helps us determine the change due to leakage.

Molar Composition

Molar composition refers to the percentage of each component in a gas mixture, expressed in terms of moles. In the exercise, the gaseous mixture consists of 97% carbon monoxide (CO) and 3% carbon dioxide (CO2). This information is essential when dealing with properties like partial pressures and total moles, but in this specific exercise, we focus mainly on the total moles. Knowing the molar composition helps ensure we perform accurate calculations and comprehend the behavior of mixed gases under different conditions.

Pressure Calculation

Pressure calculation is another critical concept in this problem. Initially, our gas mixture is at a high pressure of 138 bar. Due to a leak, the pressure drops to 129 bar. Utilizing the ideal gas law, we see how this decrease in pressure affects the number of moles in the cylinder. By plugging the values into the equation: \[ 138 \times 0.1 = n \times 0.08314 \times 303.15 \]we solve for the initial number of moles (5.47 kmol). We repeat the process with the final pressure to get the final number of moles (5.17 kmol). This shows the importance of pressure in thermodynamic calculations.

Temperature Conversion

Temperature conversion is a vital step in applying the ideal gas law, as temperatures need to be in Kelvin (K). The provided temperature in the exercise is 30°C, which must be converted to Kelvin using the conversion formula: \[ T(K) = T(°C) + 273.15 \]Thus, 30°C becomes 303.15 K. This conversion ensures consistency with the gas constant (R) units and is essential for accurate calculations. Without converting to Kelvin, the results would be incorrect, as the ideal gas law requires absolute temperature values.