How can I measure the energy contained within water vapor?

Let’s say I have a kilogram of water vapor at $\pu{423 K}$. Can I directly measure the energy, contained as heat, within the vapor? All the laws I know depend on knowing the changes that happened that led to forming this vapor at that temperature.




The absolute values of internal energy $U$ or enthalpy $H$ (which could be more useful for water vapour) at a single state point are meaningless. It is only the difference between two different state points that matter. Thus, the value for a single state point can be set to any arbitrary value. **Therefore, this arbitrary meaningless value cannot be directly measured.** Many handbooks set the arbitrary state point so that the values of these properties are positive for most liquid or gas states. For water, the triple point is selected as the reference state, where the internal energy $U$ of saturated liquid is assigned a zero value.


That's why *all the laws you know depend on knowing the changes that happened that led to forming this vapour at that temperature.* Heating a mass of $m=1\ \mathrm{kg}$ of water from the triple point with $U=0$ to a new state with $T=423\ \mathrm K$ and $p=10^5\ \mathrm{Pa}$ would give you $U=2582.7\ \mathrm{kJ}$ and $H=2776.3\ \mathrm{kJ}$.


Furthermore, you want to directly measure the energy that is *contained as heat.* You should remember that internal energy $U$ is a state quantity or state function because it describes an equilibrium state of a thermodynamic system, regardless of how the system has arrived in that state. For a closed thermodynamic system,
$$\Delta U=Q+W$$ where is $Q$ amount of heat transferred to the system and $W$ is work done on the system (according to the sign convention used by ISO and IUPAC) provided that no chemical reactions occur. Thus, simply by looking at the value of $U$, you cannot know how much was supplied by heat. You also cannot know how much heat can be extracted from it. Again *all the laws depend on knowing the changes.*


However, since the internal energy $U$ describes an equilibrium state of the thermodynamic system, regardless of how the system has arrived in that state, you do not need to know the exact path taken from the reference state to the observed state; you just need to assume a possible path. So instead of measuring $U$ directly, you can measure the values for a set of thermodynamic parameters (like mass $m$, temperature $T$, and pressure $p$ in the example above) that describe the state of the thermodynamic system. Then you can calculate the change in internal energy $\Delta U$ for a system that changes from the reference state to the new state with the observed parameter values assuming any possible path.


There are precalculated values (so-called steam tables) that already did this job for you. So you can simply measure the values for a set of thermodynamic parameters and then look up or interpolate the corresponding value for the internal energy $U$.


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