Advertisements
Advertisements
प्रश्न
Show that the slope of the p−V diagram is greater for an adiabatic process compared to an isothermal process.
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?
Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms:
| Substance | Atomic Mass (u) | Density (103 Kg m-3) |
| Carbon (diamond) | 12.01 | 2.22 |
| Gold | 197.00 | 19.32 |
| Nitrogen (liquid) | 14.01 | 1.00 |
| Lithium | 6.94 | 0.53 |
| Fluorine (liquid) | 19.00 | 1.14 |
[Hint: Assume the atoms to be ‘tightly packed’ in a solid or liquid phase, and use the known value of Avogadro’s number. You should, however, not take the actual numbers you obtain for various atomic sizes too literally. Because of the crudeness of the tight packing approximation, the results only indicate that atomic sizes are in the range of a few Å].
The specific heat capacity of water is
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 at constant temperature?
Can we define specific heat capacity for an adiabatic process?
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
Two samples A and B are initially kept in the same state. Sample A is expanded through an adiabatic process and the sample B through an isothermal process. The final volumes of the samples are the same. The final pressures in A and B are pA and pBrespectively.
Let ∆Wa and ∆Wb be the work done by the systems A and B, respectively, in the previous question.
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 sample of air weighing 1.18 g occupies 1.0 × 103 cm3 when kept at 300 K and 1.0 × 105 Pa. When 2.0 cal of heat is added to it at constant volume, its temperature increases by 1°C. Calculate the amount of heat needed to increase the temperature of air by 1°C at constant pressure if the mechanical equivalent of heat is 4.2 × 107 erg cal−1. Assume that air behaves as an ideal gas.
An ideal gas expands from 100 cm3 to 200 cm3 at a constant pressure of 2.0 × 105 Pa when 50 J of heat is supplied to it. Calculate (a) the change in internal energy of the gas (b) the number of moles in the gas if the initial temperature is 300 K (c) the molar heat capacity Cp at constant pressure and (d) the molar heat capacity Cv at constant volume.
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.
In Joly's differential steam calorimeter, 3 g of an ideal gas is contained in a rigid closed sphere at 20°C. The sphere is heated by steam at 100°C and it is found that an extra 0.095 g of steam has condensed into water as the temperature of the gas becomes constant. Calculate the specific heat capacity of the gas in J g−1 K−1. The latent heat of vaporisation of water = 540 cal g−1
4.0 g of helium occupies 22400 cm3 at STP. The specific heat capacity of helium at constant pressure is 5.0 cal K−1 mol−1. Calculate the speed of sound in helium at STP.
Standing waves of frequency 5.0 kHz are produced in a tube filled with oxygen at 300 K. The separation between the consecutive nodes is 3.3 cm. Calculate the specific heat capacities Cp and Cv of the gas.
Molar specific heat of water is C = 74.7 J/mol K, its value in cal/g K is ______.
