An adiabatic process is a thermodynamic process in which no heat is transferred between the system and its surroundings.
\[ q = 0 \]
Adiabatic conditions can be approximated by using excellent thermal insulation or by carrying out a process so rapidly that there is not enough time for significant heat transfer to occur.
Because no heat enters or leaves the system, the First Law of Thermodynamics becomes
\[ \Delta U = w \]
Any change in internal energy must therefore arise entirely from work.
Consider an ideal gas undergoing an adiabatic expansion. As the gas expands, it does work on the surroundings:
\[ w < 0 \]
Since no heat enters the system,
\[ q = 0 \]
and therefore
\[ \Delta U = w < 0 \]
The internal energy decreases as the gas performs work.
For an ideal gas, the internal energy depends only on temperature. A decrease in internal energy therefore requires a decrease in temperature:
\[ \Delta U < 0 \quad \Rightarrow \quad \Delta T < 0 \]
Consequently, an adiabatic expansion causes an ideal gas to cool. Conversely, an adiabatic compression causes the temperature to increase because work is done on the gas.
For a reversible adiabatic process,
\[ q = 0 \]
so the First Law becomes
\[ dU = dw \]
or
\[ dU = -p\,dV \]
For an ideal gas,
\[ dU = nC_V\,dT \]
giving
\[ nC_V\,dT = -p\,dV \]
Using the ideal gas law,
\[ p = \frac{nRT}{V} \]
yields
\[ nC_V\,dT = -\frac{nRT}{V}\,dV \]
or
\[ \frac{C_V}{R} \frac{dT}{T} = -\frac{dV}{V} \]
Integrating between the initial and final states gives
\[ \frac{C_V}{R} \ln \left( \frac{T_2}{T_1} \right) = -\ln \left( \frac{V_2}{V_1} \right) \]
which can be rearranged to
\[ \ln \left( \frac{T_2}{T_1} \right) = -\frac{R}{C_V} \ln \left( \frac{V_2}{V_1} \right) \]
and finally
\[ T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{R/C_V} \]
This equation allows the final temperature of a reversible adiabatic expansion or compression of an ideal gas to be calculated directly from the initial temperature and the change in volume.
Big picture: In an adiabatic process no heat is exchanged with the surroundings. Any work performed by the gas must therefore come from its internal energy. For an ideal gas this means that expansion lowers the temperature, while compression raises it.
Calculate \(\Delta T\), \(q\), \(w\), \(\Delta U\), and \(\Delta H\) for the reversible adiabatic expansion of \(1.00\ \mathrm{mol}\) of a monatomic ideal gas initially at \(375\ \mathrm{K}\), expanding from \(10.0\ \mathrm{L}\) to \(22.4\ \mathrm{L}\).
For a monatomic ideal gas,
\[ C_V = \frac{3}{2}R \]
For a reversible adiabatic expansion,
\[ T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{R/C_V} \]
\[ T_2 = 375\ \mathrm{K} \left( \frac{10.0\ \mathrm{L}} {22.4\ \mathrm{L}} \right)^{2/3} \]
\[ T_2 = 219\ \mathrm{K} \]
Therefore,
\[ \Delta T = 219\ \mathrm{K} - 375\ \mathrm{K} = -156\ \mathrm{K} \]
Because the process is adiabatic,
\[ q = 0 \]
The change in internal energy is
\[ \Delta U = nC_V\Delta T \]
\[ \Delta U = (1.00\ \mathrm{mol}) \left( \frac{3}{2} \right) \left( 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}} \right) (-156\ \mathrm{K}) \]
\[ \Delta U = -1.94\ \mathrm{kJ} \]
Since \(q=0\), the First Law gives
\[ \Delta U = w \]
\[ w = -1.94\ \mathrm{kJ} \]
Finally,
\[ \Delta H = nC_P\Delta T \]
For a monatomic ideal gas,
\[ C_P = \frac{5}{2}R \]
\[ \Delta H = (1.00\ mol) \left( \frac{5}{2} \right) (8.314 \frac{J}{mol\ K}) (-156\ K) \]
\[ \Delta H = -3.24\ \mathrm{kJ} \]
| Quantity | Value |
|---|---|
| \(\Delta T\) | \(-156\ \mathrm{K}\) |
| \(q\) | \(0\) |
| \(w\) | \(-1.94\ \mathrm{kJ}\) |
| \(\Delta U\) | \(-1.94\ \mathrm{kJ}\) |
| \(\Delta H\) | \(-3.24\ \mathrm{kJ}\) |
Physical interpretation: The gas expands and does work on the surroundings, so \(w\) is negative. Because no heat enters the system during an adiabatic expansion, the energy required to do that work comes from the internal energy of the gas, causing the temperature to decrease.
For each problem, assume a monatomic ideal gas with \(C_V=\frac{3}{2}R\) and \(C_P=\frac{5}{2}R\). Choose the correct set of values.
Big picture: Adiabatic processes exchange no heat with the surroundings. Any work performed by the system must therefore come from its internal energy. For ideal gases, this creates a direct connection between expansion, compression, and temperature change, making adiabatic pathways especially important in engines, compressors, and atmospheric processes.