# Temperature and Heat

### Written by tutor Nathan B.

Temperature and heat in physics can be understood through several approaches and each is advantageous for different purposes. Generally, these approaches fall into two categories depending on the size of the basic objects under consideration: Statistical Mechanics and Thermodynamics. Statistical Mechanics starts from atoms and builds up to the properties of larger materials, while Thermodynamics starts with the bulk properties of everyday objects. This help section emphasizes the physics of bulk materials, but provides atomic scale justifications when helpful for understanding phenomenon. To help distinguish the two approaches, all Statistical Mechanics is in blue. If you are using this material as a quick reference or basic test review, feel free to skip anything in blue. If you are reading for comprehension and depth, I highly recommend covering both approaches.

## Defining Temperature and Heat

In everyday use, temperature and heat are often synonymous with how hot or cold an object is. In physics, the three terms all serve subtly different purposes. How hot or cold an object is, or 'hotness,' refers to the total amount of kinetic energy contained in a material. In Statistical Mechanics, 'hotness' is simply the sum of all kinetic energy attributed to each atom/molecule, including the particle's velocity, rotation, and internal vibrations. Heat refers to the transfer of this kinetic energy between two systems. Since heat requires two or more systems to be defined, it is not inherent to an object. Instead, heat is specific to the context an object is placed in. Both 'hotness' and heat are quantities of energy, so their SI unit is Joules and they must be included when using energy conservation.

Finally, temperature represents the average amount of kinetic energy available to small pieces of a system. In Statistical Mechanics, temperature is more accurately defined as how many new arrangements of atoms/molecules become possible with a small increase in energy. Since an arrangement is determined by the position and velocity of all atoms, more energy usually allows many more possible velocities and, therefore, many more arrangements and a higher temperature. Because this average also depends on the type of energy available and can be complicated to calculate, temperature is usually found through its relation to other quantities, such as volume, pressure, or the color of emitted light. Temperature scales are then defined by picking two reference temperatures. The SI temperature scale is the Kelvin scale, set at 0 when all kinetic energy vanishes and approximately 273 at the freezing point of water. The more familiar Celsius and Fahrenheit scales are related to the Kelvin scale by:

K = C + 273 = 59(F - 32) + 273

where K, C, and F are temperatures in Kelvin, degrees Celsius, and degrees Fahrenheit respectively.

## Heat Transfer

In order for heat to transfer from one object to another there must be a method for the energy to be exchanged between objects. There are three main types of transfer: conduction, convection, and radiation. Conduction occurs when heat flows from one spot to another through contact without the underlying substance moving with the heat. Conduction, contrary to convection, occurs when the substance does move with the heat flow. In both cases, the kinetic energy of molecules is transferred through collisions with other molecules. In conduction, the molecules don't travel far before colliding. In convection, the molecules do travel far before colliding. Conduction is most common in solids, while convection occurs in liquids and gases. Radiation occurs when the energy in heat is converted to light energy, then absorbed by the receiving object as heat. When the sun emits light it travels through space and is then absorbed by the ground/ocean. Thus, the Earth is warmed by radiation from the Sun.

When heat is added or taken away from an object the available kinetic energy changes and, therefore, the temperature changes. Exactly how the amount of heat transferred relates to the temperature change depends on the amount and type of material being effected. Assuming the type and mass m of the material does not change, then the heat transferred Q and temperature change ΔT are related by:

Q = mcΔT

where c is a constant depending on the type of material, called the specific heat. The more heat required to raise a material's temperature by the same amount, the higher its specific heat.

Heat is a form of energy and energy is conserved; therefore, any heat transferred to an object must be lost from another object. Depending on the mass and specific heat of each object, the temperatures will change accordingly to conserve energy. Using the above equation and labeling the properties of the two objects with the index 1 or 2, then the temperature changes are related by:

m1c1ΔT1 = Q1 = -Q2 = -m2c2ΔT2

So, an object with more mass or higher specific heat will undergo a smaller temperature change than an object with less mass or lower specific heat.

The equation for specific heat assumes the material in question never changes type. This is not always true. For example, if you warm up ice in the oven it melts into water. Such drastic changes in a material are called phase changes. The most common type of phase change occurs when an object morphs between a solid, liquid, or gas; however other phase changes exist. For example, gas can turn into plasma or a metal can be cooled into a superconductor. For a phase change to occur the object must be at a specific temperature, called the critical temperature, dependent on the material and type of phase change. Furthermore, heat must be exchanged to raise or lower the material's temperature across the critical temperature. The amount of heat required for a phase change to occur will depend on the type of transition and the amount of material involved. For an object of mass m, the heat required Q for a specific phase change is given by:

Q = Lm

where L is a constant called the latent heat dependent on the type of material and transition. The more heat required per unit mass for a certain transformation to take place, the higher the latent heat of that transformation. Notice the equation of latent heat does not include a temperature change. This is because during the phase change the temperature is held at the critical temperature.

Phase changes cause dramatic changes to the structure and properties of an object at a specific temperature, but changes in temperature can also cause subtler effects. For example, as temperature increases a substance will usually expand. The amount of thermal expansion depends on the material type and initial size. For an object of initial volume V undergoing a temperature change ΔT, the change in volume ΔV is given by:

ΔV = βVΔT

where β is the average coefficient of volume expansion. Different thermal expansion coefficients explain why jar lids get stuck when left in the fridge and why cracks form in sidewalks and roads. At a molecular level, thermal expansion is the result of a larger average distance between molecules. Since molecules at a higher temperature have higher average velocities and it's easier for molecules to move apart than move closer, the particles will spend more time further away.

## Temperature and Heat Practice Quiz

The first five questions are 'easy.' They are direct applications of the definitions and relations presented without references to other areas of physics. The last five questions are 'hard.' They involve complicated algebra, knowledge of basic physics, or a particular awareness of the problem statement.

The calorie is defined as the amount of heat required to raise one gram of water by one degree Celsius. Which of the following correctly describes the unit of a calorie?

A. Energy and temperature
B. Heat and temperature
C. Energy and heat
D. Energy, heat, and temperature
The correct answer here would be C.

Given a temperature change of one Kelvin (K), one degree Celsius (C), or one degree Farhenheit (F) which of the following comparisons is true?

A. K > C > F
B. K = C > F
C. C > K > F
D. F > C = K
The correct answer here would be B.

A microwave works by emitting an electromagnetic wave (light) that is absorbed by water molecules causing them to vibrate. By what method is heat being transferred?

A. Conduction
B. Convection
C. Absorption
The correct answer here would be D.

180mL of coffee at 80°C is mixed with 15mL of creamer at 5°C. What is the final temperature of the mixture? Assume the density and specific heat of the coffee and creamer are equivalent to that of water.

A. 74°C
B. 90°C
C. 43°C
D. 20°C
The correct answer here would be A.

You have two perfectly fitting glass jars and lids at room temperature. One lid is steel and the other is aluminum. If the jars are put in a refridgerator and allowed to cool, which jar will be harder to open. The average coefficient of volume expansion for steel, aluminum, and glass is 33, 72, and 27 (x10-6/°C), respectively.

A. Steel top
B. Aluminum top
C. The lids will be easier to open
D. The lids will be equally tight
The correct answer here would be B.

You put an equal mass of pure water (c=4.2 kJ/kg°C) and salt water (c=4.0 kJ/kg°C) into separate, identical pots. The pots are placed on a stovetop and set to the same temperature. Assume heat is lost to the surroundings at a negligble rate and the liquids are originally at room temperature 20°C. Will the pure water boil (T=100°C) before the salt water (T=101°C)?

A. Yes
B. No
The correct answer here would be B.

Steam at 105°C is provided to a radiator at a steady rate. The cooled water is removed from the radiator at 90°C. What fraction of the heat transferred to the room comes directly from condensation? The specific heat of steam and water is 2.0 and 4.2 kJ/kg°C, respectively, and the latent heat of condensation from steam to water is 2,260 kJ/kg.

A. 98%
B. 67%
C. 33%
D. 2%
The correct answer here would be A.

Five ice cubes at 20g each and a temperature of 5°C are added to a liter (1kg) of soda at room temperature (20°C). The soda is in a perfectly insulated container, so no heat is lost to the surroundings. Will the ice completely melt before the system reaches a steady temperature? The specific heat of ice and soda/water is 2.1 and 4.2 kJ/kg°C, respectively, and the latent heat of melting from ice to water is 334 kJ/kg.

A. Yes
B. No
The correct answer here would be A.

For the same combination of ice and soda stated in problem eight, what temperature would the ice have to start at or below for some fraction of the ice cubes to remain in the end?

A. 0°C
B. -157°C
C. -241°C
D. -273°C
The correct answer here would be C.

A typical thermometer uses a thin, cylindrical, glass tube with a large spherical bulb at the bottom to hold mercury. Then, as the temperature rises, the mercury expands and is forced to rise up the cylindrical tube. Given the initial volume of mercury is approximately the volume of the spherical bulb (1cm3) and the cross sectional area of the tube is 3x10-4 cm2, how far will the mercury rise for a 5°C increase in temperature. The average coefficient of volume expansion for mercury is 1.8x10-4/°C and you can ignore the expansion of the glass.

A. 1cm
B. 3cm
C. 10cm
D. 30cm
The correct answer here would be B.
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