19-2 Ideal Gases#
Prompts
State the ideal gas law in terms of moles (\(n\)) and in terms of number of molecules (\(N\)). How are \(R\) and \(k\) related?
Why is the gas called “ideal”? Under what conditions do real gases approximate ideal behavior?
On a \(p\)-\(V\) diagram, what is an isotherm? For an isothermal expansion, is the work \(W\) positive or negative?
For an isothermal process, what is \(\Delta E_{\text{int}}\)? How does \(Q\) relate to \(W\)?
A gas expands at constant pressure. Write the work done by the gas in terms of \(p\) and \(\Delta V\).
Lecture Notes#
Overview#
An ideal gas obeys \(pV = nRT\) (or \(pV = NkT\))—a simple relation between pressure, volume, and temperature.
Real gases approximate ideal behavior at low density (molecules far apart, negligible interactions).
The ideal gas law links macroscopic quantities (\(p\), \(V\), \(T\)) to the amount of gas (\(n\) or \(N\)).
The ideal gas law#
In terms of moles:
\(p\): absolute pressure; \(V\): volume; \(n\): number of moles; \(T\): temperature (kelvins).
\(R = 8.31\) J/(mol·K)—gas constant.
In terms of molecules:
\(N\): number of molecules; \(k = R/N_A = 1.38 \times 10^{-23}\) J/K—Boltzmann constant.
Relation: \(nR = Nk\).
Temperature in kelvins
The ideal gas law requires absolute temperature \(T\) in kelvins. Convert from Celsius: \(T = T_C + 273.15\).
Poll: Constant-pressure heating (Charles’s law)
A container with a piston-lid contains an ideal gas at \(T = 27°\)C (300 K) and volume \(V_0\). The temperature is increased to \(127°\)C while pressure is kept constant. What is the new volume?
(A) \(V_0\)
(B) \((127/27)V_0\)
(C) \((4/3)V_0\)
(D) \((3/4)V_0\)
(E) None of these
Poll: Two rooms with different temperatures
Two identical rooms are connected by an open doorway. The temperatures in the two rooms are maintained at different values. Which room contains more air?
(A) The room with the higher temperature
(B) The room with the lower temperature
(C) The room with the higher pressure
(D) Neither (they have the same pressure)
(E) Neither (they have the same volume)
Isotherms and work#
An isotherm is a curve of constant \(T\) on a \(p\)–\(V\) diagram. For an ideal gas, \(p \propto 1/V\) along an isotherm.
Work done by the gas (general): \(W = \int_{V_i}^{V_f} p\,dV\)—area under the curve.
Process |
Work |
|---|---|
Isothermal |
\(W = nRT\,\ln(V_f/V_i)\) |
Constant volume |
\(W = 0\) |
Constant pressure |
\(W = p\,\Delta V\) |
Isothermal process: \(\Delta E_{\text{int}} = 0\) (internal energy of ideal gas depends only on \(T\)). By the first law, \(Q = W\).
Poll: Identifying process from physical setup
A person slowly pushes down on a piston, compressing a gas that is immersed in a large tub of water. Which best describes this process?
(A) Isobaric
(B) Isochoric
(C) Isothermal
Poll: Ideal gas
A fixed amount of ideal gas is compressed at constant temperature. The work \(W\) done by the gas is:
(A) Positive
(B) Negative
(C) Zero
Example: Highest temperature on a pV diagram#
Example: Highest temperature on a pV diagram
The figure shows two processes by which 1.0 g of nitrogen gas moves from state 1 to state 2. The temperature of state 1 is 27°C.
[FIGURE: pV diagram with two paths from state 1 to state 2; four labeled points (e.g., corners of a rectangle or vertices of the paths); axes labeled \(p\), \(V\)]
(a) Which of the four points has the highest temperature? (Answer without calculating all four temperatures.)
(b) Find the highest temperature among the four points.
Solution (a): For fixed \(n\), \(T \propto pV\). The point with the largest \(pV\) product has the highest temperature.
Solution (b): Use \(T = pV/(nR)\) with \(n = m/M\) for N\(_2\).
Summary#
\(pV = nRT\) or \(pV = NkT\)—ideal gas law; \(k = R/N_A\).
Isotherm: \(p \propto 1/V\) at constant \(T\).
Isothermal: \(W = nRT\,\ln(V_f/V_i)\); \(\Delta E_{\text{int}} = 0\), \(Q = W\).
Constant pressure: \(W = p\,\Delta V\).