xSkill: notes, lectures and practice for CBSE and CSE

1.1 Temperature scales Celsius, Fahrenheit and Kelvin readings are linked by linear relations. A temperature difference of one degree Celsius equals a difference of one kelvin exactly, and 1 K equals 9/5 of a Fahrenheit degree. The triple point of water is the fixed point of the modern scale, and it is assigned 273.16 K.

\frac{t _{F} - 32}{9} = \frac{t _{C}}{5} = \frac{T - 273.15}{5}, T = t_{C} + 273.15

Conversion between Fahrenheit, Celsius and absolute temperature; a difference of 1 K equals a difference of 1 degree C

1.2 Ideal gas and expansion An ideal gas obeys P V = n R T, with R = 8.31 J mol^-1 K^-1. For a fixed amount of gas, P V/T stays constant. Solids expand on heating: length, area and volume grow in proportion to the temperature change, and the three coefficients are related by beta = 2 alpha and gamma = 3 alpha. The same gamma applies to the apparent expansion of a liquid in a container, while the real expansion uses only the liquid coefficient.

P V = n R T, \frac{P V}{T} = constant

Ideal gas equation and its fixed-mass form

Δ L = Lα Δ T, Δ A = A β Δ T, Δ V = V γ Δ T, β = 2 α, γ = 3 α

Linear, areal and volume expansion of a solid, with the coefficient relations

1.3 Heat, specific heat and latent heat The heat needed to change a temperature is proportional to the mass, the specific heat capacity and the temperature change. When a substance changes state at constant temperature, the heat goes into breaking or forming bonds and is measured by the latent heat. In a calorimeter, heat lost by the hot body equals heat gained by the cold body, provided no heat leaks out.

Q = m c Δ T, Q = m L

Heat for a temperature change and heat for a change of state at constant temperature

1.4 Heat transfer Conduction carries heat through a material without bulk motion, and the rate depends on the area, the temperature gradient, the length and the thermal conductivity. Convection moves heat with a flowing fluid. Radiation transfers heat as electromagnetic waves and needs no medium; the Stefan-Boltzmann law gives the power radiated by a surface. Newton's law of cooling states that the rate of loss of heat to the surroundings is proportional to the excess temperature over the surroundings, which gives an exponential fall of temperature with time.

\frac{Q}{t} = \frac{k A ( T _{1} - T _{2} )}{L}, P = ε σ A T^{4}, σ = 5.67 \times 1 0^{- 8} W m^{- 2} K^{- 4}

Steady conduction through a slab, Stefan-Boltzmann radiated power, and the Stefan constant

\frac{d T}{d t} = - K (T - T_{s}) \Rightarrow T - T_{s} = (T_{0} - T_{s}) e^{- K t}

Newton's law of cooling and the exponential fall of temperature with time

Exam

Check the units of alpha before using it: data are often given in K^-1 or per degree C, and the two are the same for a temperature difference. Convert lengths to metres and masses to kilograms before substituting.