Systems, Heat, Work and the First Law: Quick Revision

A brief note on Systems, Heat, Work and the First Law. Read it before a test. Try each question at the end before you open its answer.

Systems

• System: the part we study. Surroundings: everything else. The boundary separates them.
• Open: energy and matter cross. Closed: only energy. Isolated: neither.
• State variables: $p$, $V$, $T$ and $n$.
• State functions ($p$, $V$, $T$, $U$) depend only on the state. Heat and work depend on the path.
• Extensive (mass, volume, $U$) grow with the amount. Intensive ($T$, $p$, density) do not.

Heat, work and U

• $U$ is the total energy stored in the system. We measure only $\Delta U$.
• Sign rule: energy into the system is positive. $q > 0$ for heat absorbed, $w > 0$ for work done on the system.
• Adiabatic: $q = 0$, so $\Delta U = w$.

The first law

• $\Delta U = q + w$. The energy of an isolated system is constant.
• Constant outside pressure: $w = -p_\text{ex}\Delta V$. $1$ L bar $= 100$ J, $1$ L atm $= 101.3$ J.
• Free expansion: $w = 0$.
• Reversible, isothermal, ideal gas: $w = -2.303\,nRT\log\dfrac{V_f}{V_i}$, the most work possible.
• Isothermal ideal gas: $\Delta U = 0$, so $q = -w$.

A worked example

A gas absorbs 500 J of heat and expands from 2.0 L to 5.0 L against a constant 1.2 bar. Find $w$ and $\Delta U$.

1. Volume change. $\Delta V = 5.0 - 2.0 = 3.0$ L.
2. Work. $w = -1.2 \times 3.0 = -3.6$ L bar $= -360$ J.
3. First law. $\Delta U = 500 + (-360) = 140$ J.

Practice with new numbers

A program to try

The program works through the example above.

Mistakes to avoid

• Dropping the minus sign in $w = -p_\text{ex}\Delta V$.
• Leaving work in L bar when $q$ is in J.
• Writing $\Delta q$ or $\Delta w$. Heat and work are not state functions.
• Thinking $q = 0$ at constant temperature. For an ideal gas it is $\Delta U$ that is zero.

Check yourself

1. Is a sealed can of soda in a fridge open, closed or isolated?

2. A gas gives out 30 J of heat and has 70 J of work done on it. What is $\Delta U$?

3. A gas expands by 0.50 L against 4.0 bar. What is $w$?

4. An ideal gas expands into a vacuum. Why is $w$ zero?