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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

The compression and rarefaction associated with a sound wave propogating in a gas are so much faster than the flow of heat in the gas that they can be treated as adiabatic processes. a) Find the speed of sound, \(v_{s}\), in an ideal gas of molar mass \(M\). b) In accord with Einstein's refinement of Newtonian mechanics, \(v_{\mathrm{s}}\) cannot exceed the speed of light in vacuum, \(c\). This fact implies a maximum temperature for an ideal gas. Find this temperature. c) Evaluate the maximum temperature of part (b) for monatomic hydrogen gas (H). d) What happens to the hydrogen at this maximum temperature?

Consider nitrogen gas, \(\mathrm{N}_{2}\), at \(20.0^{\circ} \mathrm{C}\). What is the root-mean-square speed of the nitrogen molecules? What is the most probable speed? What percentage of nitrogen molecules have a speed within \(1.00 \mathrm{~m} / \mathrm{s}\) of the most probable speed? (Hint: Assume the probability of neon atoms having speeds between \(200.00 \mathrm{~m} / \mathrm{s}\) and \(202.00 \mathrm{~m} / \mathrm{s}\) is constant. \()\)

A sample of gas at \(p=1000 . \mathrm{Pa}, V=1.00 \mathrm{~L},\) and \(T=300 . \mathrm{K}\) is confined in a cylinder. a) Find the new pressure if the volume is reduced to half of the original volume at the same temperature. b) If the temperature is raised to \(400 . \mathrm{K}\) in the process of part (a), what is the new pressure? c) If the gas is then heated to \(600 . \mathrm{K}\) from the initial value and the pressure of the gas becomes \(3000 . \mathrm{Pa},\) what is the new volume?

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 \(V_{2}\).

Compare the average kinetic energy at room temperature of a nitrogen molecule to that of a nitrogen atom. Which has the larger kinetic energy? a) nitrogen atom b) nitrogen molecule c) They have the same energy. d) It depends upon the pressure.
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