One mole of an ideal gas is heated slowly so that it goes from the PV state (P, V,) to (2P, 2V;) in such a way that the pressure is directly proportional to the volume. (a) How much work is done on the gas in the process? (Use any variable or symbol stated above along with the following as necessary: n and R.) W = (b) How is the temperature of the gas related to its volume during this process? (Do not substitute numerical values; use variables only. Use the following as necessary: n, R, V, Vi, and P.) T =

One mole of an ideal gas is heated slowly so that it goes from the PV state (P, V,) to (2P, 2V;) in such a way that the pressure is directly proportional to the volume. (a) How much work is done on the gas in the process? (Use any variable or symbol stated above along with the following as necessary: n and R.) W = (b) How is the temperature of the gas related to its volume during this process? (Do not substitute numerical values; use variables only. Use the following as necessary: n, R, V, Vi, and P.) T =

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Properties of Pure Substances

A substance with fixed chemical composition throughout is called a pure substance like water, air, and nitrogen. A pure substance does not have to be a single element or compound. A mixture of two or more phases of a pure substance is also a pure substance as long as the chemical composition is the same for all phases. These substances mainly have a constant/ uniform composition. The substances also have fixed boiling and melting points. A pure substance takes part in a chemical reaction to form predictable products.

Question

**Title: Understanding the Work Done and Temperature Relation in an Ideal Gas Process**

One mole of an ideal gas is heated slowly so that it goes from the PV state \((P_i, V_i)\) to \((2P_i, 2V_i)\) in such a way that the pressure is directly proportional to the volume.

**Concepts to Explore:**
- Work done by the gas: \(W\)
- Temperature relation during the process: \(T\)

### (a) Calculating the Work Done on the Gas in the Process

**Question:** How much work is done on the gas in the process?

[Use any variable or symbol stated above along with the following as necessary: \( n \) and \( R \).]

**Answer:**

\[ W = \]

### (b) Determining the Temperature Relation of the Gas to Its Volume During the Process

**Question:** How is the temperature of the gas related to its volume during this process?

[Do not substitute numerical values; use variables only. Use the following as necessary: \( n, R, V, V_i \) and \( P_i \).]

**Answer:**

\[ T = \]

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**Explanation of Graphs/Diagrams:**
There are no graphs or diagrams provided in the displayed image. However, the problem statement involves transitioning between states on a PV diagram where pressure and volume are directly proportional.

**Context:**
Understanding the relationship between pressure, volume, and temperature in an ideal gas is fundamental in thermodynamics. This problem helps in visualizing how work and temperature change during a thermodynamic process.

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