As the resistance of a metallic conductor is given by

R=ml/ne² tA

Where m is mass of electron, n is number of electrons, e is charge on electrons, t is relaxation time and A is area of conductor.

For a given conductor

R α 1/t

When the temperature of metal conductor is raised, the atoms or ions of the metal vibrate with greater amplitudes and greater frequencies about their mean positions. Due to increase in thermal energy, the frequency of collision of free electrons with atoms or ions while drifting towards positive end of the conductor also increases. This reduces the relaxation time. Hence, the value of resistance R increases with rise of temperature.

The resistance R_t of a metal conductor of temperature t⁰C is given by

R_t=R₀(1+αt+βt²)

Where α and β are temperature coefficients of resistance. R₀ is the resistance of conductor at 0⁰C. Their values vary from conductor to conductor.

If the temperature t⁰C is not sufficiently large, the above relation may be expressed as

R_t=R₀(1+αt)

Or           R_t=R₀+R₀ α t

Or          α= (R_t-R₀)/R₀*t

α=increase in resistance/ original resistance*rise of temp

thus temperature co-efficient of resistance is defined as the increase in resistance per unit original resistance per degree rise of temperature.

For Metals Like silver, copper etc. The value of α is positive because resistance of a metal increases with rise in temperature. The unit of α is K^-1 or C^-1

For insulators and semiconductors, α is negative, i.e the resistance decreases with rise in temperature.

For alloys like manganin, eureka and constantan, the value of α is very small as compared to that for metals. Due to high resistivity and low temperature coefficient of resistance, these alloys are used in making standard resistance coils.

The value of α is different at different temperature. Temperature coefficient of resistance averaged over the temperature range t₁ C to t₂ C is given by

α= (R₂-R₁)/R₁(t₂-t₁)