0.25 mol of a gas are compressed at a constant pressure of 250 kPa from 6000 cm3 to 2000 cm3, then expanded at a constant temperature back to 6000 cm3. What is the net work done on the gas?

Step 1: Understand the problem. The net work done on the gas involves two processes: (1) compression at constant pressure and (2) expansion at constant temperature. Work is calculated differently for each process, so we will address them separately. Step 2: Calculate the work done during the compression process. For a constant pressure process, the work done is given by the formula: W=P×(V_{f}-V_{i}), where P is the pressure, V_{i} is the initial volume, and V_{f} is the final volume. Convert the volumes from cm³ to m³ (1 cm³ = 10⁻⁶ m³) before substituting into the formula. Step 3: Calculate the work done during the expansion process. For a constant temperature process (isothermal), the work done is given by the formula: W=nRT×ln(V_{f}/V_{i}), where n is the number of moles, R is the gas constant (8.314 J/mol·K), T is the temperature, and V_{i} and V_{f} are the initial and final volumes, respectively. You will need to know the temperature to proceed with this calculation. Step 4: Determine the net work done on the gas. The net work is the sum of the work done during the compression and expansion processes. Be mindful of the signs: work done on the gas (compression) is positive, while work done by the gas (expansion) is negative. Step 5: Substitute the given values into the formulas and simplify. For compression, use P=250 kPa, V_{i}=6000 $$cm^{3}$$, and V_{f}=2000 $$cm^{3}. $$For expansion, use n=0.25, R=8.314, and the appropriate temperature value. Combine the results to find the net work.

Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 51

Chapter 19, Problem 51

0.25 mol of a gas are compressed at a constant pressure of 250 kPa from 6000 cm³ to 2000 cm³, then expanded at a constant temperature back to 6000 cm³. What is the net work done on the gas?

Verified step by step guidance

Step 1: Understand the problem. The net work done on the gas involves two processes: (1) compression at constant pressure and (2) expansion at constant temperature. Work is calculated differently for each process, so we will address them separately.

Step 2: Calculate the work done during the compression process. For a constant pressure process, the work done is given by the formula:

W = P \times (V

_f-V_i), where

P

is the pressure,

V

_i is the initial volume, and

V

_f is the final volume. Convert the volumes from cm³ to m³ (1 cm³ = 10⁻⁶ m³) before substituting into the formula.

Step 3: Calculate the work done during the expansion process. For a constant temperature process (isothermal), the work done is given by the formula:

W = n R T \times ln (V

_f/V_i), where

n

is the number of moles,

R

is the gas constant (8.314 J/mol·K),

T

is the temperature, and

V

_i and

V

_f are the initial and final volumes, respectively. You will need to know the temperature to proceed with this calculation.

Step 4: Determine the net work done on the gas. The net work is the sum of the work done during the compression and expansion processes. Be mindful of the signs: work done on the gas (compression) is positive, while work done by the gas (expansion) is negative.

Step 5: Substitute the given values into the formulas and simplify. For compression, use

P = 250 kPa

V

_i=6000 cm³, and

V

_f=2000 cm³. For expansion, use

n = 0.25

R = 8.314

, and the appropriate temperature value. Combine the results to find the net work.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Work Done on a Gas

In thermodynamics, the work done on a gas during a process can be calculated using the formula W = PΔV, where P is the pressure and ΔV is the change in volume. For processes at constant pressure, this formula simplifies the calculation of work, allowing us to determine how much energy is transferred to or from the gas as it expands or compresses.

Ideal Gas Law

The Ideal Gas Law, represented as PV = nRT, relates the pressure (P), volume (V), and temperature (T) of an ideal gas, where n is the number of moles and R is the ideal gas constant. This law is essential for understanding the behavior of gases under various conditions and can help in calculating changes in state during thermodynamic processes.

Isothermal Process

An isothermal process is a thermodynamic process that occurs at a constant temperature. During this type of process, the internal energy of an ideal gas remains constant, and any heat added to the system is used to do work. Understanding isothermal processes is crucial for analyzing the expansion phase of the gas in the given question, as it affects the work done during that stage.

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