Thermal Physics

Temperature

Temperature of a substance is a measure of the average kinetic energy of the particles comprising the substance. In layman terms, it is a measure of the hotness or coldness of an object.

Units of Temperature

There are three units of temperature in common use. These are degree Celsius, degree Fahrenheit and degree Kelvin.

A Degree Celsius ( °C ) is defined to be ( 1 ⁄ 100 ) th of the temperature difference between the boiling temperature and the freezing temperature of water. Further, the freezing temperature of water is defined to be 0 °C. Thus, the boiling temperature of water is 100°C. A degree Fahrenheit ( °F ) is defined to be ( 1 ⁄ 180 ) th of the difference between boiling temperature and freezing temperature of water. Further, the freezing temperature of water is defined to be 32°F. Hence, the boiling temperature of water is 212 °F. A unit of degree Kelvin ( °K ) is defined to be equal to a unit of degree Celsius. But degree Kelvin is defined to be zero at absolute zero temperature. Absolute zero temperature is the lowest possible temperature where motion of the particles completely ceases. Absolute temperature occures at about -273 °C. Thus, degree Kelvin can be obtained by adding 273 to degree Celsius.

T ⁄ °K = T ⁄ °C + 273

Relationship between °C and °F

For a given interval of temperature,( T - To ), The ratio between a measure in °F and a measure in °C is 9 ⁄ 5 because 100 °C is equal to 180 °F.

( T - To ) ⁄ °F = ( 9 ⁄ 5 )( T - To ) ⁄ °C

If the temperature To is taken to be the freezing temperature of water, then To ⁄ °C = 0 °C and To ⁄ °F = 32 °F.

T ⁄ °F = ( 9 ⁄ 5 )T ⁄ °C + 32

or

T ⁄ °C = ( 5 ⁄ 9 )( T ⁄ °F - 32 )

Measuring Temperature

A device used to measure temperature is called thermometer. Changes in temperature are measured in terms of physical quantities whose change varies proportionally with changes of temperature such as length of metals and volume of fluids. Let the physical quantity whose change varies proportionally with temperature be denoted by L. Then the ratio between change in temperature, ( T - To ), and the change in L, ( L - Lo ), should be a constant. Let this constant be denoted by m.

( T - To ) ⁄ ( L - Lo ) = m

The graph of T versus L should be a straight line with slope m. Let the T-intercept ( value of T when L is zero ) be denoted by b. Then the calibrating equation relating T and L should be given as follows:

T = mL + b

Callibrating a thermometer essentially means determining the values of m and b. Once the values of m and b are known the temperature can be measured by measuring L. The values of m and b can be determined from any two pairs of data relating T and L as shown in the following example.

Expansion of Metals

Substances generally expand with increase of temperature with the exception of water between 0 and 4 °C. This anomalous behaviour of water is manifested by the fact that ice floats in water and is responsible for the existence of life in water.

Linear Expansion of metals

The expansion of a metal with increase of temperature is directly proportional to the change in temperature as well as to the original length of the metal.

ΔL= αLo ΔT

ΔL = L - Lo is change in length corresponding to a change in temperature ΔT = T - To . Lo is the original length. α is a material constant called temperature coefficient of linear expansion. Its unit is 1 ⁄ °C. An expression for the length after expansion can be obtained by separating L from the equation for ΔL.

L = Lo { 1 + α( T - To ) }

Areal Expansion of Metals

Areal expansion is two dimensional expansion. Without loss of generality, let's consider the expansion of a square metal plate of length Lo. When the plate is heated, both dimensions of the plate will expand linearly. It will expand into a square plate of length Lo + ΔL = Lo + αLo ΔT. The new area is A = ( Lo + αLo ΔT )² = Lo² + 2Lo² αΔT + ( Lo αΔT )². The last term is too small to be considered because α² is much much smaller than one. Noting that the original area Ao = Lo², the following expression for the area after expansion is obtained.

A = Ao{ 1 + 2α(T - To )}

An equation relating the change in area ΔA = A - Ao and the change in temperature ΔT = T - To can be obtained by rearranging this equation.

ΔA = 2αAo ΔT

The expression 2α is called the temperature coefficient of areal expansion and denoted by γ.

Volume Expansion of Metals

Volume expansion is a three dimensional expansion. Each dimension will expand linearly. Without loss of generality, let's consider the expansion of a cube of length Lo. After being heated the cube will expand into a cube of length Lo + αLo ΔT and the new volume would be V = L 3 = ( Lo + αLo ΔT ) 3 = Lo 3 + 3Lo 3 αΔT + 3Lo 3 α 2ΔT 2 + ( Lo αΔT ) 3. The last two terms are too small to be considered because α 2 and α 3 are much much smaller than one. Noting that the original volume is Vo = Lo 3, the following expression for the volume after expansion is obtained.

V = Vo{ 1 + 3α(T - To )}

An equation between change in the volume ΔV = V - Vo and change in temperature ΔT = T - To can be obtained by rearranging this equation.

ΔV = 3αVo ΔT

The expression 3α is called the temperature coefficient of volume expansion of the material and denoted by β.