An unknown liquid has density ρ and coefficient of volume expansion β. If heat Q is added to an initial volume V and the volume increases by ΔV with no phase change, what is the liquid’s specific heat capacity c in terms of Q, ρ, β, V, and ΔV?

What links heat added to volume increase?

You are told the liquid expands when heated (no phase change), so the heat input $Q$ causes a temperature rise $\Delta T$, which in turn causes a volume change $\Delta V$. The trick is to use thermal expansion to eliminate $\Delta T$.

Relate $\Delta V$ to $\Delta T$ using volume expansion

For a liquid with coefficient of volume expansion $\beta$, $$\Delta V=\beta V\Delta T.$$ Solve for the temperature change: $$\Delta T=\frac{\Delta V}{\beta V}.$$

Use the definition of specific heat capacity

Heat added with no phase change is $$Q=mc\Delta T.$$ The mass of the initial volume $V$ is $$m=\rho V.$$ Substitute $m$ and $\Delta T$ into $Q=mc\Delta T$: $$Q=(\rho V)c\left(\frac{\Delta V}{\beta V}\right)=\rho c\frac{\Delta V}{\beta}.$$ Now solve for $c$: $$c=\frac{Q\beta}{\rho\Delta V}.$$

Quick unit check

$\beta$ has units $1/\text{K}$, $\rho\Delta V$ has units of kg, so $Q\beta/(\rho\Delta V)$ gives J/(kg·K), which matches specific heat capacity.