Howell AP Physics B Notes

Thermal Physics Mechanical Equivalent of Heat

James Prescott Joule, a British physicist, experimentally determined that thermal energy and mechanical energy are related.

Heat is the exchange of thermal energy between systems at different temperatures.

If a change in temperature occurs due to a mechanical process, the amount of thermal energy gained or lost is equal to the amount of mechanical energy lost or gained.

The Law of Conservation of Energy states that energy cannot be created or destroyed only transferred from one system to another or transformed from one type from another.

Mechanical energy is one form of energy while thermal energy is another. Friction can transform mechanical energy into thermal energy by heating the surfaces in contact.

Heat Transfer

Conduction: transfer of thermal energy from one system to another through physical contact.

Conductors are materials that allow thermal energy to flow easily.

Insulators are materials that do not allow thermal energy to flow easily.

The amount of thermal energy transferred or the amount of heat can be calculated:

• is the rate for thermal energy transfer,

• is the thermal conductivity of a conductor. It is specific for each material, in

• is the cross-sectional area of the conductor, • is the change in temperature,

• is the length of the conductor,

Convection: as a fluid increases temperature, it expands. The body of fluid then rises, carrying the thermal energy with it. The surrounding lower temperature fluid falls, is heated then rises. Convection occurs due to uneven heating of fluids that in turn move the thermal energy.

Radiation: thermal energy transfer through electromagnetic waves. Electromagnetic waves are any waves in the electromagnetic spectrum; visible light, infrared light and ultra-violet light.

All matter emits radiation.

The higher the temperature of object, the higher the frequency of the radiation emitted.

Thermal Expansion

When heated or cooled, most materials expand or contract respectively.

The amount of expansion, for most materials, is a linear function

• is the change in length, . • is the coefficient of expansion, • is the initial length, • is the change in temperature, K.

Similar equations can be derived for both area and volume expansions Kinetic Theory

Howell AP Physics B Notes

Ideal gas: does not truly exist in nature, but dilute gases are very close approximations of ideal gas behavior. An ideal gas behaves according to the ideal gas law equation:

• is the pressure of the gas, • is the volume of the gas, • is the number of moles of gas,

• is the universal gas constant. It is equivalent to NAk, (Avogadro’s number

times Boltzman’s constant) =

• is the temperature, .

The ideal gas law can also be written as

• is the number of molecules

• is Boltzman’s constant

For a closed container, where n is held constant, the ideal gas law can be written as a ratio

Since n is held constant and R is a constant, nR can be eliminated from the equation, resulting in

the combined gas law equation.

Due to the movement of the molecules of an ideal gas, each molecule has kinetic energy. The

average kinetic energy of all molecules determines the temperature of the gas, the higher temperature the faster molecules move. Within the container the molecules collide, pressure is due to collisions between the molecules and the walls of the container.

The average kinetic energy of particles is directly related to the temperature of the gas and follows the following equation:

• is the average kinetic energy of the particles,

• is Boltzman’s constant

• is the temperature, measured in

The average speed of a particle is

The mass of a molecule is the molar mass, M, divided by Avogadro’s number, NA.

Howell AP Physics B Notes

• is the root-mean-square velocity,

• is the universal gas constant. =

• is the temperature, .

• is the molar mass, .

The sum of all the individual energies of the molecules is the total internal energy of the gas, U.

• is the internal energy, • is the number of moles of gas,

• is the universal gas constant. =

• is the temperature,

The ideal gas law is , which when compared to , gives

Laws of Thermodynamics

Zeroeth Law: Thermal equilibrium is reached when two or more objects that are in thermal contact have the same temperature.

Thermal contact simply means thermal energy can be exchanged between the two objects; they do not have to be in physical contact.

First Law: The change in a gas’s internal energy is dependent on the thermal energy added to or removed from the gas and the work done on the gas

• is the change in internal energy of a gas,

• is the thermal energy added to (or removed from) the gas, • is the work done on the gas,

Second Law: Thermal energy naturally flows from the warmer object to the cooler object, never in the reverse direction.

Third Law: There is no temperature lower than absolute zero, and absolute zero is unattainable. PV Diagrams and Thermodynamic Processes

A PV diagram is a graph of pressure versus volume.

A line draw between two points on the graph can be describe by a thermodynamic process, where thermal energy is transferred, work is done, or both.

A constant pressure (isobaric) process is one in which the pressure is held constant while the volume changes, as shown in Figure 8.1.

Howell AP Physics B Notes

The volume can increase, shown by the right arrow, or decrease, shown by the left arrow.

When the volume increases, the gas expands, pushing on the environment.

A push is a force. Therefore the gas is applying a force through a displacement, doing work.

Work is force x distance.

Pressure is force per area.

Force is pressure x area

Area x distance is volume

Therefore, work done by the gas is pressure x change in volume

The work done on the gas is

• is the work done by the environment on the gas, • is the pressure of the gas, • is the change in volume of the gas, measured in

For a PV diagram, the area under the curve is the work done on or by the gas, as shown in Figure 8.2

Howell AP Physics B Notes

A constant volume (or isochoric, isovolumetric or isometric) process is one in which the volume is held constant and the pressure changes, as shown in Figure 8.3

In an isochoric process, the first law of thermodynamics simplifies to

The thermal energy added to or removed from the gas is equal to the change in internal energy

of the gas.

An isothermal process is one in which the temperature does not change, thus nRT in the ideal gas law equation is constant, meaning the PV cannot change. An isotherm is in the shape of a hyperbola, as shown in Figure 8.4

In an isothermal process, the first law of thermodynamics simplifies to

The thermal energy added to the gas is equal to the negative work done on the gas (the gas does

positive work on the environment as the gas expands).

An adiabatic process is one in which no thermal energy is exchanged with the gas, . An adiabat is similar in shape to an isotherm but must descend more steeply than a hyperbola, shown in Figure 8.5

Howell AP Physics B Notes

The first law of thermodynamics for an adiabatic process simplifies to

The change in internal energy is equal to the work done on the gas.

Heat Engines and Efficiency

In general, efficiency is the ratio of work done by a system to the energy put into the system.

• is the efficiency, no units • is the work done by a system, • is the thermal energy added to the system,

A heat engine is a device to transform thermal energy into work, such as a steam engine.

Water is heated to steam, the steam pushes a piston and cools off, the cool water is pumped back to the boiler and the process starts again.

The maximum theoretical efficiency of a heat engine depends on the temperatures of the two reservoirs.

The work done is the difference in the thermal energy added to the system in the hot reservoir, QH, and the thermal energy left in the cold reservoir, QC, after the work is done.

Efficiency is the work done divided by the thermal energy supplied,

• is the efficiency • is the work done, • QH is the thermal energy transferred from the hot reservoir,

• is the thermal energy transferred to the cold reservoir,

Carnot engine is a theoretical heat engine that operates at maximum efficiency between two temperatures, and the processes are completely reversible.

The efficiency of a Carnot engine depends on the temperature, , of the environment from which the thermal energy is transferred to the engine, and the temperature, , of the environment to which the thermal energy is transferred from the engine, . Therefore

This proportionality can be substituted into the efficiency equation

• is the efficiency • is the cold reservoir temperature,

Howell AP Physics B Notes

• is the hot reservoir temperature, .

The Carnot engine cycle consists of four processes, as shown in Figure 8.6, two isothermal and two adiabatic processes.

• Process A to B = isothermal (∆U = 0) • Process B to C = adiabatic (Q=0) • Process C to D = isothermal (∆U = 0) • Process D to A = adiabatic (Q=0) Cyclic Processes

The work done by a gas during a cyclic process is the area enclosed by the function on the graph.

A “clock-wise” process in one in which net positive work is done by the gas, system, as shown in Figure 8.7a.

A “counter-clockwise” process is one in which net negative work is done by the gas, as shown in Figure 8.7b.