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
Given below are observations on molar specific heats at room temperature of some common gases.
| Gas |
Molar specific heat (Cv) (cal mol–1 K–1) |
| Hydrogen | 4.87 |
| Nitrogen | 4.97 |
| Oxygen | 5.02 |
| Nitric oxide | 4.99 |
| Carbon monoxide | 5.01 |
| Chlorine | 6.17 |
The measured molar specific heats of these gases are markedly different from those for monatomic gases. Typically, molar specific heat of a monatomic gas is 2.92 cal/mol K. Explain this difference. What can you infer from the somewhat larger (than the rest) value for chlorine?
A geyser heats water flowing at the rate of 3.0 litres per minute from 27 °C to 77 °C. If the geyser operates on a gas burner, what is the rate of consumption of the fuel if its heat of combustion is 4.0 × 104 J/g?
Calculate the mass of ice required to lower the temperature of 300 g of water 40°C to water at 0°C.
(Specific latent heat of ice = 336 J/g, the Specific heat capacity of water = 4.2J/g°C)
What do you understand by the following statements:
The heat capacity of the body is 60JK-1.
A refrigerator converts 100 g of water at 20°C to ice at -10°C in 35 minutes. Calculate the average rate of heat extraction in terms of watts.
Given: Specific heat capacity of ice = 2.1 J g-1°C-1
Specific heat capacity of water = 4.2 J g-1°C-1
Specific latent heat of fusion of ice = 336 J g-1
A liquid X has the maximum specific heat capacity and is used as a coolant in Car Radiators. Name the liquid X.
Heat supplied to a solid change it into liquid. What is this change in the phase called?
Two metallic blocks P and Q of masses in ratio 2: 1 are given the same amount of heat. If their temperature rise by the same amount, compare their specific heat capacities.
Why do the farmers fill their fields with water on a cold winter night?
A heater of power P watt raises the temperature of m kg of a liquid by Δt K in time t s. Express
the specific heat capacity of liquid in terms of above data.
Why do bottled soft drinks get cooled, more quickly by the ice cubes than by the iced water, both at 0℃?
How does green house effect help in keeping the temperature of earth’s surface suitable for living of human beings?
Give three reasons for the increase of green house gases.
How will rise in sea level affect population in coastal countries?
How will climate changes affect the various animal species?
The global warming has resulted:
(a) the increase in yield of crops
(b) the decrease in sea levels
(c) the decrease in human deaths
(d) the increase in sea levels
(i) State whether the specific heat capacity of a substance remains the same when its state changes from solid to liquid.
(ii) Give one example to support your answer.
What are the factors on which the quantity of heat given to a body depends?
Write two advantages of high specific heat capacity of water.
A certain amount of heat Q will warm 1 g of material X by 3°C and 1 g of material Y by 4°C. Which material has a higher specific heat capacity?
A vessel of negligible heat*capacity contains 40g of ice in it at 0°C, 8g of steam at 100°C is passed into the ice to melt it. Find the final temperature of the contents of the vessel.
(Specific latent heat of vaporization of steam = 2268 J/g, specific latent heat of fusion of ice = 336 J/f and specific heat capacity of water = 4.2 J/g°C)
Calculate the amount of heat released when 5.0 g of water at 20°C is changed into ice at 0°C.
(Specific heat capacity of water = 4.2 J/g°C
Specific latent heat of fusion of ice = 336 J/g)
From the options given below the specific heat of _______ is maximum.
The cold object the hot object enclosed in one box of heat-resistant material.
- What changes will occur in the two objects when temperature flows from those objects?
- Which principle can show that the energy exchange takes place between two objects only when kept in isolated system?
What is specific heat capacity?
Why is water used as a coolant in radiators of a car?
The molar specific heats of an ideal gas at constant pressure and volume are denoted by Cp and Cv, respectively. If `gamma = "C"_"p"/"C"_"v"` and R is the universal gas constant, then Cv is equal to ______.
In the method of mixtures, a hot metal is dropped into cold water and the mixture reaches a final temperature. Which principle is used to find the specific heat of the metal?
