# 3.3 First law of thermodynamics  (Page 4/6)

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## Vaporizing water

When 1.00 g of water at $100\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ changes from the liquid to the gas phase at atmospheric pressure, its change in volume is $1.67\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}\text{.}$ (a) How much heat must be added to vaporize the water? (b) How much work is done by the water against the atmosphere in its expansion? (c) What is the change in the internal energy of the water?

## Strategy

We can first figure out how much heat is needed from the latent heat of vaporization of the water. From the volume change, we can calculate the work done from $W=p\text{Δ}V$ because the pressure is constant. Then, the first law of thermodynamics provides us with the change in the internal energy.

## Solution

1. With ${L}_{v}$ representing the latent heat of vaporization, the heat required to vaporize the water is
$Q=m{L}_{v}=\left(1.00\phantom{\rule{0.2em}{0ex}}\text{g}\right)\left(2.26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{J/g}\right)=2.26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{J}.$
2. Since the pressure on the system is constant at $1.00\phantom{\rule{0.2em}{0ex}}\text{atm}=1.01\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}{\phantom{\rule{0.2em}{0ex}}\text{N/m}}^{2}$ , the work done by the water as it is vaporized is
$W=p\text{Δ}V=\left(1.01\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}{\phantom{\rule{0.2em}{0ex}}\text{N/m}}^{2}\right)\left(1.67\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}\right)=169\phantom{\rule{0.2em}{0ex}}\text{J}\text{.}$
3. From the first law, the thermal energy of the water during its vaporization changes by
$\text{Δ}{E}_{\text{int}}=Q-W=2.26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{J}-169\phantom{\rule{0.2em}{0ex}}\text{J}=2.09\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{J}\text{.}$

## Significance

We note that in part (c), we see a change in internal energy, yet there is no change in temperature. Ideal gases that are not undergoing phase changes have the internal energy proportional to temperature. Internal energy in general is the sum of all energy in the system.

Check Your Understanding When 1.00 g of ammonia boils at atmospheric pressure and $-33.0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C,}$ its volume changes from 1.47 to $1130{\phantom{\rule{0.2em}{0ex}}\text{cm}}^{3}$ . Its heat of vaporization at this pressure is $1.37\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{J/kg}\text{.}$ What is the change in the internal energy of the ammonia when it vaporizes?

$1.26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{J}\text{.}$

View this site to learn about how the first law of thermodynamics. First, pump some heavy species molecules into the chamber. Then, play around by doing work (pushing the wall to the right where the person is located) to see how the internal energy changes (as seen by temperature). Then, look at how heat added changes the internal energy. Finally, you can set a parameter constant such as temperature and see what happens when you do work to keep the temperature constant ( Note: You might see a change in these variables initially if you are moving around quickly in the simulation, but ultimately, this value will return to its equilibrium value).

## Summary

• The internal energy of a thermodynamic system is a function of state and thus is unique for every equilibrium state of the system.
• The increase in the internal energy of the thermodynamic system is given by the heat added to the system less the work done by the system in any thermodynamics process.

## Conceptual questions

What does the first law of thermodynamics tell us about the energy of the universe?

Does adding heat to a system always increase its internal energy?

If more work is done on the system than heat added, the internal energy of the system will actually decrease.

A great deal of effort, time, and money has been spent in the quest for a so-called perpetual-motion machine, which is defined as a hypothetical machine that operates or produces useful work indefinitely and/or a hypothetical machine that produces more work or energy than it consumes. Explain, in terms of the first law of thermodynamics, why or why not such a machine is likely to be constructed.

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