Advertisements
Advertisements
प्रश्न
Show that the slope of the p−V diagram is greater for an adiabatic process compared to an isothermal process.
Advertisements
उत्तर
In an isothermal process,
PV = k ...(i)
On differentiating it w.r.t V, we get
`"V" (dP)/(dV) + "P" = 0`
`(dP)/(dV) = -"P"/"V"`
`(dP)/(dV) = - "k"/"V"^2` [ Using (i)] , k = constant
k = constant
In an adiabatic process,
PVγ = K ...(ii)
On differentiating it w.r.t V, we get
`"V"^gamma(d"P")/(d"V")+ gamma"PV"^(gamma-1) = 0`
`(d"P")/(d"V") = -( gamma "P""V"^(gamma-1))/"V" ^ (gamma+1) ["Using" (ii) , γ > 1 ]` and
K is constant
`gamma and (d"P")/(d"V")`
are the slope of the curve and the ratio of heat capacities at constant pressure and volume, respectively; P is pressure and V is volume of the system.
By comparing the two slopes and keeping in mind that γ >1 , we can see that the slope of the P-V diagram is greater for an adiabatic process than an isothermal process.
APPEARS IN
संबंधित प्रश्न
A metre long narrow bore held horizontally (and closed at one end) contains a 76 cm long mercury thread, which traps a 15 cm column of air. What happens if the tube is held vertically with the open end at the bottom?
Does a gas have just two specific heat capacities or more than two? Is the number of specific heat capacities of a gas countable?
Can we define specific heat capacity for an adiabatic process?
Does a solid also have two kinds of molar heat capacities Cp and Cv? If yes, is Cp > Cv? Or is Cp − Cv = R?
In a real gas, the internal energy depends on temperature and also on volume. The energy increases when the gas expands isothermally. Examining the derivation of Cp − Cv = R, find whether Cp − Cv will be more than R, less than R or equal to R for a real gas.
Can a process on an ideal gas be both adiabatic and isothermal?
Can two states of an ideal gas be connected by an isothermal process as well as an adiabatic process?
In an isothermal process on an ideal gas, the pressure increases by 0.5%. The volume decreases by about
Consider the processes A and B shown in the figure. It is possible that
5 g of a gas is contained in a rigid container and is heated from 15°C to 25°C. Specific heat capacity of the gas at constant volume is 0.172 cal g−1 °C−1 and the mechanical equivalent of heat is 4.2 J cal−1. Calculate the change in the internal energy of the gas
A mixture contains 1 mole of helium (Cp = 2.5 R, Cv = 1.5 R) and 1 mole of hydrogen (Cp= 3.5 R, Cv = 2.5 R). Calculate the values of Cp, Cv and γ for the mixture.
The figure shows two vessels with adiabatic walls, one containing 0.1 g of helium (γ = 1.67, M = 4 g mol−1) and the other containing some amount of hydrogen (γ = 1.4, M = 2 g mol−1). Initially, the temperatures of the two gases are equal. The gases are electrically heated for some time during which equal amounts of heat are given to the two gases. It is found that the temperatures rise through the same amount in the two vessels. Calculate the mass of hydrogen.
An engine takes in 5 moles of air at 20°C and 1 atm, and compresses it adiabatically to `1/10^"th"` of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be X kJ. The value of X to the nearest integer is ______.
If at same temperature and pressure, the densities for two diatomic gases are respectively d1 and d2 then the ratio of velocities of sound in these gases will be ______.
