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Chapter 19: Problem 20

A relationship that gives the pressure, \(p\), of a substance as a function of its density, \(\rho\), and temperature, \(T\), is called an equation of state. For a gas with molar mass \(M\), write the Ideal Gas Law as an equation of state.

Short Answer

Expert verified
Answer: The equation of state of an ideal gas in terms of density (蟻), temperature (T), and molar mass (M) is: p = 蟻RT/M

Step by step solution

01

Write down the Ideal Gas Law

The Ideal Gas Law is given by: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
02

Substitute the number of moles using mass and molar mass

We know that n (number of moles) can be written as the mass (m) divided by the molar mass (M). Substituting m/M for n, we get: PV = (m/M)RT
03

Rewrite mass in terms of density and volume

Since density (蟻) is defined as mass (m) divided by volume (V), we can represent mass as the product of density and volume, that is m = 蟻V. Substituting 蟻V for m, we get: PV = (蟻V/M)RT
04

Simplify the equation

Notice that the volume (V) appears on both sides of the equation. We can divide both sides by volume to isolate the pressure (P): P = 蟻RT/M
05

Final equation of state

The Ideal Gas Law in terms of density (蟻), pressure (p), temperature (T), and molar mass (M) is: p = 蟻RT/M

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Most popular questions from this chapter

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 \(V_{2}\).

One mole of an ideal gas, at a temperature of \(0^{\circ} \mathrm{C}\), is confined to a volume of \(1.0 \mathrm{~L}\). The pressure of this gas is a) \(1.0 \mathrm{~atm}\). b) 22.4 atm. c) \(1 / 22.4 \mathrm{~atm}\) d) \(11.2 \mathrm{~atm}\).

Interstellar space far from any stars is usually filled with atomic hydrogen (H) at a density of 1 atom/cm \(^{3}\) and a very low temperature of \(2.73 \mathrm{~K}\). a) Determine the pressure in interstellar space. b) What is the root-mean-square speed of the atoms? c) What would be the edge length of a cube that would contain atoms with a total of \(1.00 \mathrm{~J}\) of energy?

Consider a box filled with an ideal gas. The box undergoes a sudden free expansion from \(V_{1}\) to \(V_{2}\). Which of the following correctly describes this process? a) Work done by the gas during the expansion is equal to \(n R T \ln \left(V_{2} / V_{1}\right)\) b) Heat is added to the box. c) Final temperature equals initial temperature times \(\left(V_{2} / V_{1}\right)\). d) The internal energy of the gas remains constant.

Chapter 13 examined the variation of pressure with altitude in the Earth's atmosphere, assuming constant temperature-a model known as the isothermal atmosphere. A better approximation is to treat the pressure variations with altitude as adiabatic. Assume that air can be treated as a diatomic ideal gas with effective molar mass \(M_{\text {air }}=28.97 \mathrm{~g} / \mathrm{mol}\) a) Find the air pressure and temperature of the atmosphere as functions of altitude. Let the pressure at sea level be \(p_{0}=101.0 \mathrm{kPa}\) and the temperature at sea level be \(20.0^{\circ} \mathrm{C}\) b) Determine the altitude at which the air pressure and density are half their sea-level values. What is the temperature at this altitude, in this model? c) Compare these results with the isothermal model of Chapter \(13 .\)
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