Advertisements
Advertisements
प्रश्न
Prove the Mayer's relation `C_p - C _v = R/J`
Advertisements
उत्तर
Consider one mole of an ideal gas that is enclosed in a cylinder by a light, frictionless, airtight piston.
Let P, V, and T be the pressure volume and temperature of the gas, respectively. If the gas is heated so that its temperature rises by dT, but volume remains constant.
Let dQ1 → be the amount of heat supplied to the gas,
dE → be the increase in internal energy. Since, volume of gas is constant, no work is done in moving the piston.
dQ1 = dE + dW
∴ dQ1 = dE = CVdT ...(i)
(As dW = 0)
If the gas is heated to the same temperature at constant pressure.
Let dV → be the increase in volume.
Therefore work is done and it is given by
dW = PdV
Let dQ2 → be the heat supplied to the gas
∴ dQ2 = dE + dW
∴ CpdT = dE + dW ...(ii)
Putting dE = CvdT and dW = PdV
∴ Equation (ii) becomes
CPdT = CvdT + PdV
(Cp − Cv)dT = PdV ...(iii)
For one mole of gas
PV = RT
∴ pdV = RdT
∴ (Cp − Cv)dT = RdT
Cp − Cv = R ...(iv)
This is known as Mayer's relation.
Generally, heat supplied is measured in calories, and work done is measured in joules. Therefore, equation (iv) is modified as follows:.
`C_p - C _v = R/J`
APPEARS IN
संबंधित प्रश्न
A solid of mass 50 g at 150 °C is placed in 100 g of water at 11 °C when the final temperature recorded is 20 °C. Find the specific heat capacity of the solid. (specific heat capacity of water = 4.2 J/g °C)
Heat energy is supplied at a constant rate to 100g of ice at 0 °C. The ice is converted into water at 0° C in 2 minutes. How much time will be required to raise the temperature of water from 0 °C to 20 °C? [Given: sp. heat capacity of water = 4.2 J g-1 °C-1, sp. latent heat of ice = 336 J g-1].
Name three fossil fuels that emit carbon dioxide into the atmosphere ?
How will global warming disturb the ecological balance?
How will you prove experimentally that different substances have different specific heat capacities?
The ratio of specific heat capacity to molar heat capacity of a body _____________ .
Give one example where high specific heat capacity of water is used as cooling.
Specific heat capacity of a substance A is 3.8 J g-1 K-1 and of substance B is 0.4 J g-1 k-1. Which substance is a good conductor of heat? How did you arrive at your conclusion?
A liquid X has specific heat capacity higher than the liquid Y. Which liquid is useful as coolant in car radiators.
If, in a central heating system, steam enters a radiation pipe at 100°C and water leaves the radiation pipe at 100°C, can this radiation pipe heat a room? Give an explanation for your answer.
Write an expression for the heat energy liberated by a hot body.
1 kg of water freezes to form ice at 0°C. What amount of heat is withdrawn?
Discuss how high specific heat capacity of water helps in formation of land and sea breeze.
A piece of ice is heated at a constant rate. The variation of temperature with heat input is shown in the graph below:
(i) What are represented by AB and CD?
(ii) What conclusion can you draw regarding the 110°c nature of ice from the above graph?
Derive an expression for finding out the specific heat capacity of a body (solid) from the readings of an experiment given below:
(i) Mass of empty calorimeter (with stirrer) = m1 gm
(ii) Mass of the metal piece = M gm
(iii) Mass of colorimeter and water = m2 gm
(iv) Initial temperature and water = t1°C
(v) Temperature of hot solid (metal piece) = t2 °C
(vi) Final temperature of the mixture = t°C
(vii) Specific heat of calorimeter = 0.4 J gm / °C
How much heat energy is necessary to raise the temperature of 5 kg of water from 20°C to 100°C?
The SI unit of specific heat is _______.
_______ is defined as the amount of heat required to raise the temperature of 1kg of a substance by 1°C.
The heat capacity of the vessel of mass 100 kg is 8000 J/°K. Find its specific heat capacity.
What is specific heat capacity?
Numerical Problem.
What is the heat in joules required to raise the temperature of 25 grams of water from 0°C to 100°C? What is the heat in Calories? (Specific heat of water = `(4.18"J")/("g"°"C")`
A diatomic gas undergoes adiabatic change. Its pressure 'P' and temperature 'T' are related as p ∝ Tx, where x is ______.
The ratio of the specific heats `c_"p"/c_"v"=gamma` in terms of degrees of freedom 'n' is given by ______.
The diagram below shows a cooling curve for 200 g of water. The heat is extracted at the rate of 100 Js-1. Answer the questions that follow:
- Calculate specific heat capacity of water.
- Heat released in the region BC.
Two metals A and B have specific heat capacities in the ratio 2:3. If they are supplied same amount of heat then
If specific heat capacity of metal A is 0.26 Jg-1 0C-1 then calculate the specific heat capacity of metal B.
Two cylinders of equal height and radius are made of copper and aluminum. Which of them conducts heat faster?
What is the amount of heat required to raise the temperature of 5 kg of iron from 30°C to 130°C? Specific heat capacity of iron = 483 Jkg-1C-1.
A block of ice of mass 120 g at temperature 0°C is put in 300 gm of water at 25°C. The xg of ice melts as the temperature of the water reaches 0°C. The value of x is ______.
[Use: Specific heat capacity of water = 4200 Jkg-1K-1, Latent heat of ice = 3.5 × 105 Jkg-1]
