Mixed problems: pistons, springs, gravity

In more general problems the gas can push weights, springs or other bodies. In every case the first law is the same: every form of energy (kinetic energy of a weight, elastic energy of a spring, gravitational potential energy, thermal energy of a second body) enters the balance.

The typical set-up is a vertical cylinder with a piston of mass $m_p$ and area $A$. The gas pressure is constant because the piston is in mechanical equilibrium at every instant: the weight of the piston adds to the push of the atmosphere.

$P_\text{gas} = P_\text{atm} + \frac{m_p\,g}{A}$

When the gas receives heat and lifts the piston by $\Delta h$, the energy balance of the gas includes three “outflows” for work:

• lifting the piston: $\Delta E_\text{pot,pistone} = m_p\,g\,\Delta h$;
• work against the atmosphere: $P_\text{atm}\,A\,\Delta h$;
• the increase in internal energy of the gas: $\Delta E_\text{int} = \frac{f}{2}nR\,\Delta T$.

Balance with a heavy piston

$\ev{Q = \Delta E_\text{int} + m_p\,g\,\Delta h + P_\text{atm}\,A\,\Delta h}$ with the geometric constraint $\Delta V_\text{gas} = A\,\Delta h$ and the equation of state $P_\text{gas}V_\text{gas} = nRT_\text{gas}$.

The two auxiliary conditions (geometric constraint and equation of state) close the system: they allow $\Delta h$, $\Delta T$ and $\Delta V$ to be linked and hence the required unknown to be solved for.

In this chapter we have seen that energy conservation alone (with the work of the atmosphere included) allows a great variety of irreversible thermodynamic problems to be solved: the gas expands abruptly, two bodies reach thermal equilibrium, the piston rises — all without ever imagining a path on the $P$-$V$ diagram. But questions remain that the first law cannot answer: why does heat always flow from hot to cold? Which transformations are reversible? To answer these a new quantity is needed, entropy.

A quantitative case (a vertical piston being lifted) is worked out in Lifting a vertical piston.

Links

Topics: Thermodynamics Concepts: First law of thermodynamics · Internal energy · Gravitational potential energy Skills: Bubble energy balance Objects: Ideal gas · Piston and cylinder

Related exercises: Nitrogen heated in a rigid cylinder · Lifting a vertical piston · Heating under the piston