Question

A cylinder fitted with a piston contains \(5.00 \mathrm{~L}\) of a gas at a pressure of \(4.00 \mathrm{~atm} .\) The entire apparatus is contained in a water bath to maintain a constant temperature of \(25^{\circ} \mathrm{C}\). The piston is released and the gas expands until the pressure inside the cylinder equals the atmospheric pressure outside, which is 1 atm. Assume ideal gas behavior and calculate the amount of work done by the gas as it expands at constant temperature.

Short answer

Given that the problem does not provide or request the numerical value for the number of moles of the gas \(n\), the answer would be expressed in terms of \(n\) as \(W = -n \times 24.68 \text{ J mol}^{-1}\).

Step-by-step solution

Understand the nature of the process

Since the expansion occurs at a constant temperature (isothermal), we will use the formula for work done by an ideal gas during isothermal expansion or compression, which is given by\( W = -nRT \times \text{ln}\frac{V_f}{V_i} \), where \( R \) is the ideal gas constant, \( n \) is the number of moles of gas, \( T \) is the temperature in Kelvin, \( V_i \) is the initial volume, and \( V_f \) is the final volume. The negative sign indicates work done by the system.

Convert the given temperature to Kelvin

The temperature must be in Kelvin for the ideal gas calculations. To convert from Celsius to Kelvin, use the formula \( T(K) = T(^\text{o}C) + 273.15 \). For this scenario, the temperature is \(25^\text{o}C\), so convert that to Kelvin: \( T = 25 + 273.15 = 298.15~K \).

Calculate the final volume using the ideal gas law

We use the ideal gas law \( P_i V_i = P_f V_f \) since \(n\) and \(R\) are constant during the isothermal process. We know the initial pressure \(P_i = 4 \text{ atm}\), the initial volume \(V_i = 5.00 \text{ L}\), and the final pressure \(P_f = 1 \text{ atm}\). By rearranging the ideal gas law, we find the final volume \( V_f \): \(V_f = \frac{P_i V_i}{P_f} = \frac{4 \text{ atm} \times 5.00 \text{ L}}{1 \text{ atm}} = 20.00 \text{ L}\).

Calculate work done using the isothermal work formula

Now, substitute the initial and final volumes into the isothermal work equation. However, we do not know the moles \(n\), but \(n\) will cancel out because it's the same gas at an initial and final state and the temperature is constant. Thus, \(W = -nRT \times \text{ln}\frac{V_f}{V_i} = -nRT \times \text{ln}\frac{20.00}{5.00} = -nRT \times \text{ln}(4)\). We can now insert the value for \(T\) and the gas constant \(R = 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1}\) to find the work done \(W\): \(W = -n \times 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \times 298.15 \text{ K} \times \text{ln}(4)\).

Simplify and solve for the work done

The natural log of 4 is approximately 1.3863. We now calculate \(W\): \(W = -n \times 0.0821 \times 298.15 \times 1.3863\). Note that \(n\) was never given, but let's assume that we knew \(n\) or it is included in our unit of work since the exercise is asking only for the work without specifying further details. The result of the expression gives the work in units of liter-atm; to convert it to joules, we can use the conversion factor \(1 \text{ L atm} = 101.325 \text{ J}\). The work done by the gas as it expands at constant temperature can now be reported in joules.

Key concepts

Ideal Gas Law

The ideal gas law is a crucial equation in thermodynamics that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas using the equation PV = nRT. An ideal gas is a hypothetical gas composed of non-interacting point particles that perfectly fit this relationship. This equation can be used to predict the behavior of real gases under many conditions, although it breaks down at very high pressures or very low temperatures where real gases deviate from ideal behavior.

To apply the ideal gas law, we must make sure all quantities are in appropriate units, where pressure is usually measured in atmospheres (atm), volume in liters (L), temperature in Kelvin (K), and the amount of substance is in moles (mol). In our exercise, we used the ideal gas law to calculate the final volume of the gas after expansion by holding temperature and moles constant and adjusting for the change in pressure.

Work Done by Gas

When we talk about work in the context of thermodynamics, we are often referring to the work done by or on a system, in this case, a gas. For isothermal processes, where the temperature remains constant, the work done by an ideal gas during expansion or compression can be expressed by the formula: \( W = -nRT \times \ln\frac{V_f}{V_i} \). The negative sign indicates work done by the gas on its surroundings. The work done depends on the initial and final volumes, the number of moles of gas, the ideal gas constant, and the absolute temperature.

During an isothermal expansion, gas particles do work as they push against an external pressure, and this formula allows us to calculate how much energy is transferred in that process. If the volume increases, as in our exercise, work is done by the gas as it pushes the piston outward, and we consider this work to be positive. Isothermal expansion requires that the system is in thermal equilibrium with a heat reservoir, meaning that heat flows into the system to maintain the constant temperature while the work is being done.

Ideal Gas Constant

The ideal gas constant, denoted as R, is a proportionality factor that appears in the ideal gas law equation, bridging the units of pressure, volume, temperature, and mole quantity. It has a fixed value of \( R = 0.0821 \text{ L atm K}^{-1} \text{mol}^{-1} \) when pressure is in atmospheres, volume is in liters, and temperature in Kelvin. This constant is key to making calculations involving the behavior of gases.

The ideal gas constant is derived from fundamental physical constants and is the same for all ideal gases. It is a crucial component in our exercise to determine the amount of work done during the isothermal process. By knowing the value of R, along with other properties like temperature and change in volume, it's possible to predict the work involved in gas expansions or compressions, making it an invaluable tool for scientists and engineers working in fields ranging from meteorology to mechanical engineering.