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Chapter 19: Problem 73

Suppose 5.0 moles of an ideal monatomic gas expand at a constant temperature of \(22^{\circ} \mathrm{C}\) from an initial volume of \(2.0 \mathrm{~m}^{3}\) to \(8.0 \mathrm{~m}^{3}\) a) How much work is done by the gas? b) What is the final pressure of the gas?

Short Answer

Expert verified
Answer: The work done by the gas during the isothermal expansion is approximately 9988.05 Joules, and the final pressure of the gas after the expansion is approximately 1531.18 Pascal.

Step by step solution

01

Identify the values given in the problem

For this problem, you have been given: 1. Initial volume (V1) = 2.0 m³ 2. Final volume (V2) = 8.0 m³ 3. Number of moles (n) = 5.0 moles 4. Temperature (T) = 22°C (we need to convert this to Kelvin)
02

Convert the temperature to Kelvin

The Celsius temperature should be converted to Kelvin, which can be done using the following formula: T(K) = T(°C) + 273.15 So, T(K) = 22 + 273.15 = 295.15 K
03

Calculate the work done during the isothermal expansion

Using the formula of work done during an isothermal expansion: W = nRT * ln(V2/V1) We need to find the value of R (Universal gas constant) which is in the appropriate units. Here, the units are moles, Kelvin, and m³. So, we use the value of R = 8.314 J/(mol*K). Now, plug in the values: W = (5.0 moles) * (8.314 J/(mol*K)) * (295.15 K) * ln(8.0 m³ / 2.0 m³) W ≈ 9988.05 J Therefore, the work done by the gas during the isothermal expansion is approximately 9988.05 Joules.
04

Calculate the initial pressure of the gas

Use the ideal gas law to find the initial pressure of the gas. Rearrange the ideal gas law formula to solve for the initial pressure (P1): P1 = nRT / V1 Plugging in the values: P1 = (5.0 moles) * (8.314 J/(mol*K)) * (295.15 K) / (2.0 m³) P1 ≈ 6124.73 Pa The initial pressure of the gas is approximately 6124.73 Pascal.
05

Calculate the final pressure of the gas after the expansion

We can use the ideal gas law again to find the final pressure (P2): P2 = nRT / V2 Plugging in the values: P2 = (5.0 moles) * (8.314 J/(mol*K)) * (295.15 K) / (8.0 m³) P2 ≈ 1531.18 Pa Therefore, the final pressure of the gas after the isothermal expansion is approximately 1531.18 Pascal.

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Most popular questions from this chapter

1 .00 mol of molecular nitrogen gas expands in volume very quickly, so no heat is exchanged with the environment during the process. If the volume increases from \(1.00 \mathrm{~L}\) to \(1.50 \mathrm{~L},\) determine the work done on the environment if the gas's temperature dropped from \(22.0^{\circ} \mathrm{C}\) to \(18.0^{\circ} \mathrm{C}\). Assume ideal gas behavior.

A monatomic ideal gas expands isothermally from \(\left\\{p_{1}, V_{1}, T_{1}\right\\}\) to \(\left\\{p_{2}, V_{2}, T_{1}\right\\} .\) Then it undergoes an isochoric process, which takes it from \(\left\\{p_{2}, V_{2}, T_{1}\right\\}\) to \(\left\\{p_{1}, V_{2}, T_{2}\right\\}\) Finally the gas undergoes an isobaric compression, which takes it back to \(\left\\{p_{1}, V_{1}, T_{1}\right\\}\) a) Use the First Law of Thermodynamics to find \(Q\) for each of these processes. b) Write an expression for total \(Q\) in terms of \(p_{1}, p_{2}, V_{1},\) and \(V_{2}\).

At Party City, you purchase a helium-filled balloon with a diameter of \(40.0 \mathrm{~cm}\) at \(20.0^{\circ} \mathrm{C}\) and at \(1.00 \mathrm{~atm} .\) a) How many helium atoms are inside the balloon? b) What is the average kinetic energy of the atoms? c) What is the root-mean-square speed of the atoms?

At a temperature of \(295 . \mathrm{K},\) the vapor pressure of pentane \(\left(\mathrm{C}_{5} \mathrm{H}_{12}\right)\) is \(60.7 \mathrm{kPa}\). Suppose \(1.000 \mathrm{~g}\) of gaseous pentane is contained in a cylinder with diathermal (thermally conducting) walls and a piston to vary the volume. The initial volume is \(1.000 \mathrm{~L},\) and the piston is moved in slowly, keeping the temperature at \(295 \mathrm{~K}\). At what volume will the first drop of liquid pentane appear?

Air in a diesel engine cylinder is quickly compressed from an initial temperature of \(20.0^{\circ} \mathrm{C}\), an initial pressure of \(1.00 \mathrm{~atm}\), and an initial volume of \(600 . \mathrm{cm}^{3}\) to a final volume of \(45.0 \mathrm{~cm}^{3}\). Assuming the air to be an ideal diatomic gas, find the final temperature and pressure.
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