Question

Methane is heated in a rigid container from \(80 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\) to \(300^{\circ} \mathrm{C}\). Determine the final pressure of the methane treating it as \((a)\) an ideal gas and \((b)\) a Benedict-WebbRubin gas.

Short answer

1 = 293.15\ K$) - Final temperature: \(T_2 = 40^\circ C\) (convert this to Kelvin by adding 273.15, $T_2 = 313.15\ K$) #Part a: Final pressure using the Ideal Gas Law# Using the ideal gas law, we have: \[ PV = nRT, \] where: - \(P\) = pressure (in kPa) - \(V\) = volume (in L) - \(n\) = moles - \(R\) = ideal gas constant (\(8.314\ kJ/(kmol\cdot K)\)) - \(T\) = temperature (in K) Knowing that the initial and final volume remains constant in the process, we get the relation: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ Now, we can find the final pressure \(P_2\) as follows: $P_2 = P_1 \cdot \frac{T_2}{T_1} = 80\ kPa \cdot \frac{313.15\ K}{293.15\ K} = 85.07\ kPa$ #Part b: Final pressure using the Benedict-Webb-Rubin Equation of State# Using the Benedict-Webb-Rubin (BWR) equation of state, the equation is $P = \frac{RT}{v - b} - \frac{a}{v(v + c)}$ The constants a, b, and c are specific to methane and given as: - \(a = 0.427\) - \(b = 0.0866\) - \(c = 0.114\) Calculate the molar volume \(v\) by dividing the volume by the moles, using the ideal gas law: $V = \frac{nRT}{P_1} => v = \frac{RT_1}{P_1}$ Substitute values: $v = \frac{8.314 \cdot 293.15}{80} = 30.46 \frac{L}{mol}$ Now, evaluate the BWR equation at the initial state: $P_1 = \frac{RT_1}{v-b} - \frac{a}{v(v + c)}$ and at the final state: $P_2 = \frac{RT_2}{v-b} - \frac{a}{v(v + c)}$ No values are given for the vaporized portion to use the BWR equation appropriately. Hence, the solution cannot be determined using this equation. #Summary# In summary, the final pressure using the Ideal gas law is \(85.07\ kPa\). The final pressure using the Benedict-Webb-Rubin equation of state cannot be determined due to insufficient data about the vaporized portion of methane.

Step-by-step solution

Initial Data

Given data: - Initial pressure: \(P_1 = 80\ kPa\) - Initial temperature: \(T_1 = 20^\circ C\) (convert this to Kelvin by adding 273.15, $T_

Key concepts

Ideal Gas Law

Understanding the behavior of gases under different conditions is crucial in the field of thermodynamics. The ideal gas law is a fundamental concept which provides a rough approximation for the behavior of most gases under a variety of conditions. It is expressed by the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) is the volume, \( n \) indicates the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin.

When applying the ideal gas law to our problem where methane is heated in a rigid container, the change in temperature is key to finding the new pressure. Since the volume \( V \) remains constant in a rigid container, and the quantity of gas (number of moles \( n \) ) does not change, any increase in temperature \( T \) will result in a proportional increase in pressure \( P \). By rearranging the ideal gas law to solve for the final pressure \( P_2 \), we can simply calculate it using the initial pressure \( P_1 \) and temperatures \( T_1 \) and \( T_2 \).

• First, convert the initial temperature from degrees Celsius to Kelvin by adding 273.15.
• Then, use the ideal gas law to solve for the final pressure after noting the final temperature in Kelvin.

Benedict-Webb-Rubin Equation

Going beyond the simplicity of the ideal gas law, real gases exhibit behaviors that deviate due to intermolecular forces and the actual volume occupied by the gas molecules. The Benedict-Webb-Rubin (BWR) equation is an advanced thermodynamic equation that accounts for these real gas effects. It includes additional parameters over the ideal gas law, thus providing a more accurate description of gas behavior under a wide range of temperatures and pressures.

Using the BWR equation requires specific constants unique to each gas, which must be determined experimentally, and it can be quite complex. Nevertheless, for methane or any real gas, the BWR equation calculates the pressure by including these factors and more closely matches real-world results compared to the ideal gas law. If students grasp the ideal gas law and can engage with the BWR equation as an extension, they will significantly deepen their understanding of thermodynamics of gases.

Rigid Container Process

A key aspect of this exercise involves heating methane in a 'rigid container.' This term describes a scenario in thermodynamics where the volume of the container does not change regardless of the internal pressure or temperature. This is known as an isochoric process.

Understanding rigid container processes is essential for students, as it greatly simplifies calculations. In our exercise, because the volume remains unchanged, we can ignore the variable \( V \) when using the ideal gas law or the BWR equation. This assumption points students toward focusing solely on the relationship between temperature and pressure, which are directly related as one increases or decreases, so does the other, assuming mass and volume are constant. Noting this, the increase in temperature as methane is heated should immediately suggest an increase in pressure within the container.

Temperature-Pressure Relationship

The temperature-pressure relationship, often called Gay-Lussac's Law in the context of an isochoric process, is the principle that describes how gases' pressure increases as temperature increases when volume is held constant.

In our methane example, the container restrains the volume, so as the temperature goes up, the molecules inside the container move faster, collide more frequently and with greater force against the container walls, leading to an increase in pressure. Our previous equations, the ideal gas law and the BWR equation, are rooted in this fundamental understanding. By ensuring students understand this simple but powerful concept, we lay the groundwork for them to tackle more complex thermodynamic problems involving relationships between pressure, volume, and temperature.