A certain ideal gas has a molar specific heat of CV=(7)/(2) R A 2.00 -mol sample of the gas alwa

A certain ideal gas has a molar specific heat of CV=(7)/(2)...

A certain ideal gas has a molar specific heat of $C_{V}=\frac{7}{2} R$ A 2.00 -mol sample of the gas always starts at pressure $1.00 \times 10^{5} \mathrm{Pa}$ and temperature $300 \mathrm{K}$. For each of the following processes, determine (a) the final pressure, (b) the final volume, (c) the final temperature, (d) the change in internal energy of the gas, (e) the energy added to the gas by heat, and (f) the work done on the gas. (i) The gas is heated at constant pressure to $400 \mathrm{K}$. (ii) The gas is heated at constant volume to $400 \mathrm{K}$. (iii) The gas is compressed at constant temperature to $1.20 \times 10^{5}$ Pa. (iv) The gas is compressed adiabatically to $1.20 \times 10^{5} \mathrm{Pa}$.

A certain ideal gas has a molar specific heat of $C_{V}=\frac{7}{2} R$
A 2.00 -mol sample of the gas always starts at pressure $1.00 \times 10^{5} \mathrm{Pa}$ and temperature $300 \mathrm{K}$. For each of the following processes, determine (a) the final pressure, (b) the final volume, (c) the final temperature, (d) the change in internal energy of the gas, (e) the energy added to the gas by heat, and (f) the work done on the gas. (i) The gas is heated at constant pressure to $400 \mathrm{K}$. (ii) The gas is heated at constant volume to $400 \mathrm{K}$. (iii) The gas is compressed at constant temperature to $1.20 \times 10^{5}$ Pa. (iv) The gas is compressed adiabatically to $1.20 \times 10^{5} \mathrm{Pa}$.

Key Concepts

Isothermal Process in Ideal Gases

In an isothermal process, the temperature remains constant, implying that the internal energy of the ideal gas does not change (?U = 0). As a result, any work done by or on the system is exactly balanced by heat transfer. This process leverages the ideal gas law to relate changes in volume and pressure at constant temperature and is fundamental for understanding systems in thermal equilibrium.

Process-Specific Thermodynamic Analysis

Different thermodynamic processes, such as constant pressure, constant volume, isothermal, and adiabatic processes, have distinct relationships that govern the behavior of the system. Each process dictates how variables like pressure, volume, temperature, and internal energy change, and they require tailored methods to calculate work, heat transfer, and changes in state.

Adiabatic Process in Ideal Gases

An adiabatic process occurs without heat exchange (Q = 0) between the system and its surroundings. For ideal gases, this process is characterized by the relation PV^? = constant, where ? (gamma) is the ratio of specific heats (Cp/Cv). This relationship is crucial when determining how pressure and volume will change during rapid compressions or expansions where no heat is transferred.

Specific Heat Capacities

Specific heat capacities, such as molar heat capacity at constant volume (Cv) and at constant pressure (Cp), describe the amount of heat required to raise the temperature of one mole of a gas by one degree under constant volume or pressure conditions. For ideal gases, these are used to calculate the change in internal energy and to relate the energy changes in different thermodynamic processes, with their relation given by Cp = Cv + R.

First Law of Thermodynamics

The first law of thermodynamics, expressed as ?U = Q - W, establishes the conservation of energy within a thermodynamic system by equating the change in internal energy (?U) to the net heat added (Q) minus the work done by the system (W). This principle underpins all energy analyses in thermodynamic processes.

Ideal Gas Law

The ideal gas law, PV = nRT, is fundamental for relating pressure, volume, and temperature in any thermodynamic process involving an ideal gas. It is used to determine unknown state variables when the other quantities (n, R, T, and one of P or V) are known, making it essential for analyzing various process conditions.