"If two systems are separately in thermal equilibrium with a third system, they are also in thermal equilibrium with each other".
Zeroth Law of Thermodynamics
Consider two systems in thermal contact, with their boundaries touching. We know that either something happens or nothing happens. Measure all the properties of both systems, then leave them for as long as necessary before measuring again. If none of the properties have changed, nothing has happened and the systems are in thermal equilibrium.
Imagine an array of ten billion cylinders. They all have the same pressure and volume at the beginning, but different pressures and volumes at the end. There is, however, one property for which they all have the same value. This property has the same value for all systems in thermal equilibrium with each other, and it determines thermal equilibrium. That property is temperature.
The temperature of a body is a measure of the amount of heat energy it possesses. Temperature (T or θ) is normally measured in degrees Celsius (°C); this scale is based on the freezing and boiling points of water, which are 0°C and 100°C respectively. However, the SI unit of temperature is the degree Kelvin (°K or K); this is also the unit of absolute temperature, so 0K is absolute zero. Heat is generated or transferred by the random motion of molecules, atoms or electrons, and the kinetic energy of the molecules of a gas is directly proportional to absolute temperature. The spacing of degrees on the Kelvin scale is the same as that on the Celsius scale, and 0K≈-273°C.
If something does happen between the two systems, heat energy flows from the hotter system to the colder one, lowering the temperature of the former and raising that of the latter, until the temperatures of the systems are equal. This happens regardless of boundaries. Disregarding heat lost to the surroundings, the heat lost by the hotter system must equal that gained by the colder one.
The change in the heat energy possessed by a system is given by the formula ΔQ=mcΔθ. c denotes the specific heat capacity, which is numerically equal to the energy needed to raise the temperature of 1kg of a substance by 1K. Its units are J/kgK. Applying this equation to the two systems gives m1c1(T1-TE)=m2c2(TE-T2), where 1 and 2 denote the hotter and colder systems respectively and TE denotes the equilibrium temperature.
The temperature of a body is measured using a thermometer. The temperature is normally shown by the height of a column of liquid, but any physical property may be utilised to make a thermometer. Examples are the potential difference (voltage) between two metals in contact, the pressure of a gas whose volume is constant, and the volume of a gas whose pressure is constant.
The act of measuring should not change the value of the property being measured. From the equations for the change in heat energy, it can be seen that the change in temperature of a body decreases with increasing mass. Where the thermometer uses a column of liquid, to take an example, it follows that the volume of that liquid must be much smaller than the volume of the body whose temperature is being measured.
Ice and water can coexist at 0°C, and water and steam can coexist at 100°C. These facts can be used to calibrate a thermometer. First, insert the thermometer into a large tank of ice and water. The thermometer will settle at a particular point, which is taken to correspond to 0°C. Then insert the thermometer into a tank of water and steam. Its reading will change again, and the new point at which it settles is taken to correspond to 100°C.
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