# 3.2 Work, heat, and internal energy  (Page 3/6)

 Page 3 / 6
${E}_{\text{int}}=\sum _{i}\left({\stackrel{\text{−}}{K}}_{i}+{\stackrel{\text{−}}{U}}_{i}\right),$

where the summation is over all the molecules of the system, and the bars over K and U indicate average values. The kinetic energy ${K}_{i}$ of an individual molecule includes contributions due to its rotation and vibration, as well as its translational energy ${m}_{i}{v}_{i}^{2}\text{/}2,$ where ${v}_{i}$ is the molecule’s speed measured relative to the center of mass of the system. The potential energy ${U}_{i}$ is associated only with the interactions between molecule i and the other molecules of the system. In fact, neither the system’s location nor its motion is of any consequence as far as the internal energy is concerned. The internal energy of the system is not affected by moving it from the basement to the roof of a 100-story building or by placing it on a moving train.

In an ideal monatomic gas, each molecule is a single atom. Consequently, there is no rotational or vibrational kinetic energy and ${K}_{i}={m}_{i}{v}_{i}^{2}\text{/}2$ . Furthermore, there are no interatomic interactions (collisions notwithstanding), so ${U}_{i}=\text{constant}$ , which we set to zero. The internal energy is therefore due to translational kinetic energy only and

${E}_{\text{int}}=\sum _{i}{\stackrel{\text{−}}{K}}_{i}=\sum _{i}\frac{1}{2}{m}_{i}\overline{{v}_{i}^{2}}.$

From the discussion in the preceding chapter, we know that the average kinetic energy of a molecule in an ideal monatomic gas is

$\frac{1}{2}{m}_{i}\stackrel{\text{−}}{{v}_{i}^{2}}=\frac{3}{2}{k}_{\text{B}}T,$

where T is the Kelvin temperature of the gas. Consequently, the average mechanical energy per molecule of an ideal monatomic gas is also $3{k}_{\text{B}}T\text{/}2,$ that is,

$\overline{{K}_{i}+{U}_{i}}=\stackrel{\text{−}}{{K}_{i}}=\frac{3}{2}{k}_{\text{B}}T.$

The internal energy is just the number of molecules multiplied by the average mechanical energy per molecule. Thus for n moles of an ideal monatomic gas,

${E}_{\text{int}}=n{N}_{\text{A}}\left(\frac{3}{2}{k}_{\text{B}}T\right)=\frac{3}{2}nRT.$

Notice that the internal energy of a given quantity of an ideal monatomic gas depends on just the temperature and is completely independent of the pressure and volume of the gas. For other systems, the internal energy cannot be expressed so simply. However, an increase in internal energy can often be associated with an increase in temperature.

We know from the zeroth law of thermodynamics that when two systems are placed in thermal contact, they eventually reach thermal equilibrium, at which point they are at the same temperature. As an example, suppose we mix two monatomic ideal gases. Now, the energy per molecule of an ideal monatomic gas is proportional to its temperature. Thus, when the two gases are mixed, the molecules of the hotter gas must lose energy and the molecules of the colder gas must gain energy. This continues until thermal equilibrium is reached, at which point, the temperature, and therefore the average translational kinetic energy per molecule, is the same for both gases. The approach to equilibrium for real systems is somewhat more complicated than for an ideal monatomic gas. Nevertheless, we can still say that energy is exchanged between the systems until their temperatures are the same.

## Summary

• Positive (negative) work is done by a thermodynamic system when it expands (contracts) under an external pressure.
• Heat is the energy transferred between two objects (or two parts of a system) because of a temperature difference.
• Internal energy of a thermodynamic system is its total mechanical energy.

A proton initially at rest falls through a p.d of 25000V. what speed does it gain?
what is the reaction of heat on magnet
what is a physical significant of electric dipole moment .
A dipole moment it's a mechanical electrical effect used in nature
Antonio
what is the uses of carbon brushes in generator
at what temperature is the degree Fahrenheit equal to degree Celsius
Celsius and Faharaneith are different, never equal
Antonio
find their liners express of n=a+b/T² ( plot graph n against T)
Radio Stations often advertis "instant news,,if that meens you can hear the news the instant the radio announcer speaks it is the claim true? what approximate time interval is required for a message to travel from Cairo to Aswan by radio waves (500km) (Assume the waves Casbe detected at this range )
what is growth and decay
thevenin
Can someone please predict the trajectory of a point charge in a uniform electric field????
what is deference between strong force and coulomb force
so basically the electric lines or the field lines
it's the measure of the flow of an electric field through a certain area
What is electric flux
how to calculate the magnetic force if the B field is depend on the coordinate