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ChemistryEdu Logo Thermodynamics | Heat, Work and Internal Energy#

Heat (q)#

  • Heat is a form of energy, which flows due to difference in temperature.
  • It flows from high temperature to low temperature.
  • Heat is a path function, ie, it depends on the path followed by the system to reach to the current state.
  • Sign convention: Heat absorbed by the system is taken as positive and heat released by the system is taken as negative.

Heat exchange

Work (W)#

  • Work is a mode of transfer of energy. Work can be done either through expansion or compression.

  • Work is a path function, ie, it depends on the actual path followed by the system to reach the current state.

  • Sign convention: Work done by the system is negative and work done on the system is positive.

  • In expansion, work is done by the system, so work done is negative. In compression, work is done on the system, so work done is positive.

Work done by system

  • Work done by the system is calculated as:
\[dW = -Fdx\]

Here, we have negative sign because direction of external force, F and displacement dx is opposite.

\[Since,\ F = Pressure \times Area\]
\[dW = -PAdx\]
\[dW = -PdV\]
\[Integrating\ both\ sides:\]
\[\int dW = -\int\limits_{V_1}^{V_2} PdV\]
\[W = -\int\limits_{V_1}^{V_2} PdV\]

Here, since P is the external pressure, we can say that work done by the system is:

\[W = -\int\limits_{V_1}^{V_2} P_{ext}dV\]
  • Since the geometrical meaning of integration is finding area under curve, so we can also calculate work done by finding area under P-V curve (for a reversible process in which Pext is approximately equal to Pgas). We will discuss reversible process later.

Internal Energy (U)#

  • The sum of all possible kinds of energy present in the system is called its internal energy.
  • Thus, internal energy constitutes kinetic energy, potential energy, electronic energy, nuclear energy, rotational energy, vibrational energy etc.
  • Internal energy is a state function, ie, it depends only on initial and final state of the system.
  • For a cyclic process, ΔU = 0
  • It is an extensive property, ie, it depends on amount of matter present in the system.

Calculation of internal energy#

  • Internal energy can be expressed as a function of volume and temperature.
\[U = f(V,T)\]
\[dU = {({dU \over dV})}_TdV + {({dU \over dT})}_VdT\]
\[where,\ {({dU \over dV})}_T = Change\ in\ U\ w.r.t\ V\ at\ constant\ T\]
\[and,\ {({dU \over dT})}_V = Change\ in\ U\ w.r.t\ T\ at\ constant\ V\]
  • For an ideal gas, volume cannot be changed at constant temperature because there is no force of attraction or repulsion existing between two ideal gas molecules.
\[So,\ {({dU \over dV})}_T = 0\]
  • Thus, for an ideal gas, we can write:
\[dU = {({dU \over dV})}_T dT\]
  • For one mole of ideal gas:
\[C_v = {({dU \over dV})}_T\]
\[Hence,\ dU = C_vdT\]
  • For n moles of ideal gas:
\[dU = nC_vdT\]
\[Integrating\ both\ sides:\]
\[ΔU = \int\limits_{T_1}^{T_2} nC_vdT\]
\[If\ C_v\ is\ independent\ of\ temperature:\]
\[ΔU = nC_v(T_2 - T_1)\]
\[where,\ C_v = Molar\ heat\ capacity\ at\ constant\ volume\]

Note

  • For a cyclic process, ΔU = 0.
  • For an isothermal process, ΔU = 0 (Since, T2 = T1).