First Law of Thermodynamics

Heat as energy was not physical law until it was demonstrated by James Prescott Joule. In tying a descending mass to a paddle wheel immersed in water, and measuring the increased temperature of the water as the mass dropped, Joule solidified the first law of thermodynamics: heat and work are forms of energy, and energy is conserved.

Mathematically, the first law can be written as

\[\Delta U=\Delta Q+\Delta W,\]

where \(\Delta U\) is the internal energy of the system, \(\Delta Q\) is the heat supplied to the system, and \(\Delta W\) is the work done to the system.

A typical thermodynamic system [1]

A system is the object under investigation. All the things attempting to interact with the system are called surroundings. The boundary is the region that separates the system and the surroundings. For example, if an egg is boiling, the inside of the egg is the system, the pot of water is the surroundings, and the shell is the boundary.

An open system allows both energy and matter to be transferred through the boundary.

A building with an open front door, such as a school or an open shop, is an open system. In this case, both matter (the people entering and exiting; the air flowing in and out) and energy (heat from the sunlight passing through the windows) are allowed to permeate the boundary.

A closed system does not allow the transfer of matter, but does allow the transfer of energy through the boundary.

A sealed can of soda is a closed system. While matter in the forms of liquid and air cannot get through the sealed metal of the can, it is possible for energy in the form of heat to be transferred from inside the can to the surroundings, as occurs when the can is left in a refrigerator for an extended period of time.

An isolated system is highly concealed so that neither energy nor matter can be transferred through the boundary.(see the figure).

While a perfectly isolated system is very difficult to construct in practice, a closed thermos is an example of a device that tries. So long as the lid is shut on the thermos, certainly no matter is allowed to enter or exit. Furthermore, the entire design of thermos is meant to obstruct the flow of heat into or out of the contents inside.

Heat capacity (C) is the amount of heat required to raise the temperature of an object by 1 K. It can be given intensively either per mole (molar heat capacity) or per unit mass (specific heat capacity). Extensively, it is given by

\[C=\dfrac{Q}{\Delta T}.\]

Initially, two rooms are thermally isolated from each other and their surroundings. The heat capacity of room A is 100 quanta of energy per Kelvin. The heat capacitance of room B is 70 quanta of energy per Kelvin.

When the two rooms are thermally connected and allowed to reach equilibrium, what is the ratio of the change in temperature in room A to that of room B, \(\dfrac{\Delta T_A}{\Delta T_B}\)?

---

Since the rooms are thermally isolated from their surroundings, the heat gained by one will equal the heat gained by the other:

\[Q_A= -Q_B.\]

Solving the heat capacity equation for Q gives

\[Q=C\Delta T.\]

So,

\[C_A \Delta T_A = -C_B \Delta T_B \implies \dfrac{\Delta T_A}{\Delta T_B} = -\dfrac{C_B}{C_A} = -\dfrac{70}{100} = -0.7.\]

Specific Heat Capacity

This section requires expansion.

\(dU=TdS-pdV\)

This section requires expansion.

[1] Image from https://commons.m.wikimedia.org/wiki/File:System_boundary2.svg under the creative Commons license for reuse and modification.