Question

An amount of 1 mole of a gas is changed from its initial state \((20 \mathrm{~L}, 2 \mathrm{~atm})\) to final state (4L, \(10 \mathrm{~atm}\) ), respectively. If the change can be represented by a straight line in \(P-V\) curve, the maximum temperature achieved by the gas in the process is \((R=0.08 \mathrm{~L}-\mathrm{atm} / \mathrm{K}-\mathrm{mol})\) (a) \(900^{\circ} \mathrm{C}\) (b) \(900 \mathrm{~K}\) (c) \(627 \mathrm{~K}\) (d) \(1173^{\circ} \mathrm{C}\)

Short answer

The maximum temperature achieved by the gas in the process is 5000 K.

Step-by-step solution

State the Ideal Gas Law

The relation between the pressure (P), volume (V), number of moles (n), ideal gas constant (R), and temperature (T) for an ideal gas is given by the Ideal Gas Law: PV = nRT. We will use this equation to find the maximum temperature achieved by the gas.

Determine the slope of the P-V curve

Since the change in the state of the gas can be represented by a straight line on the P-V curve, we can determine the slope of the line, which represents (dP/dV). The slope is calculated by the change in pressure divided by the change in volume between the initial and final states: slope = (P_final - P_initial) / (V_final - V_initial).

Calculate the slope

Substitute the initial and final pressures and volumes into the slope formula: slope = (10 atm - 2 atm) / (4 L - 20 L) = 8 atm / (-16 L) = -0.5 atm/L.

Apply the Ideal Gas Law at the initial state

Apply the Ideal Gas Law using the initial state information: P_initial * V_initial = n * R * T_initial, where P_initial = 2 atm, V_initial = 20 L, n = 1 mole and R = 0.08 L-atm/K-mol.

Calculate the initial temperature

Solve for T_initial: T_initial = (P_initial * V_initial) / (n * R) = (2 atm * 20 L) / (1 mole * 0.08 L-atm/K-mol) = 40 K.

Determine the expression for temperature at any state

Rewrite the Ideal Gas Law to express temperature as a function of pressure and volume: T = PV / (nR).

Calculate the temperature for the final state

Substitute the final state values into the temperature expression: T_final = P_final * V_final / (nR) = 10 atm * 4 L / (1 mole * 0.08 L-atm/K-mol) = 400 / 0.08 K = 5000 K.

Determine the maximum temperature

The maximum temperature corresponds to the highest value of PV along the line connecting the initial and final states. On the P-V curve, T = PV/(nR) should be at its maximum at the highest value of PV, which we calculated to be during the final state. Therefore, the maximum temperature achieved by the gas in the process is 5000 K.

Key concepts

Understanding the Ideal Gas Law (PV=nRT)

The Ideal Gas Law, represented by the equation \(PV = nRT\), is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas in terms of pressure (\(P\)), volume (\(V\)), number of moles (\(n\)), and temperature (\(T\)). The constant \(R\) is known as the Ideal Gas Constant. This equation allows us to calculate the state of a gas when any three of its state variables are known.

The problem discussed requires applying this law to find the maximum temperature achieved by a gas. Remember that for any calculation using this formula, consistency in units is crucial. For example, pressure should typically be in atmospheres (atm), volume in liters (L), temperature in Kelvin (K), and the ideal gas constant should correspond to those units, which is \(0.08 \text{L-atm/K-mol}\) in this case.

The Ideal Gas Constant (R)

The Ideal Gas Constant (\(R\)) is a proportionality constant in the Ideal Gas Law equation and its value depends on the units used for pressure, volume, and temperature. For instance, when using atmospheres for pressure, liters for volume, and Kelvin for temperature, the value of \(R\) is \(0.0821 \text{L-atm/K-mol}\), or \(8.314 \text{J/K-mol}\) when using SI units (Pascals for pressure and cubic meters for volume). This constant is derived from empirical measurements and is essential to ensure that the equation has consistent units and gives accurate predictions of gas behavior.

The correct value of \(R\) must be used for calculations to be valid. This is why we used \(0.08 \text{L-atm/K-mol}\) according to the problem's given units.

Slope of the P-V Curve

In the context of a \(P-V\) curve, which graphically represents the relationship between the pressure and volume of a gas, the slope of the curve tells us how pressure changes with volume. A straight-line path on the \(P-V\) curve implies that the process is linear, with a constant ratio between pressure change and volume change. The slope is negative for an ideal gas when pressure decreases with increasing volume, or positive when pressure increases with decreasing volume.

The slope of the line is calculated by \(\text{slope} = (\Delta P/\Delta V)\), which denotes the change in pressure to the change in volume. The exercise provided shows a decrease in volume with an increase in pressure, resulting in a negative slope (\( -0.5 \text{atm/L}\)), indicative of a compression process.

Temperature Calculation in Gases

Knowing how to calculate the temperature of a gas is crucial for understanding its thermal state. Temperature in gas laws is always measured in Kelvin, which is an absolute temperature scale. To convert Celsius to Kelvin, one must add 273.15 to the Celsius value.

In the given problem, temperature calculations are performed at the initial and final states of the gas's transformation using the Ideal Gas Law. By rearranging the equation, we get \(T = \frac{PV}{nR}\). Be mindful that when dealing with temperatures in gas laws, it's essential to ensure they are in the proper units (Kelvin) to avoid discrepancies in calculations. The maximum temperature in the given problem was calculated to be 5000 K. Remember that temperature directly reflects the average kinetic energy of the gas molecules.