Advertisements
Advertisements
प्रश्न
An amount n (in moles) of a monatomic gas at an initial temperature T0 is enclosed in a cylindrical vessel fitted with a light piston. The surrounding air has a temperature Ts (> T0) and the atmospheric pressure is Pα. Heat may be conducted between the surrounding and the gas through the bottom of the cylinder. The bottom has a surface area A, thickness x and thermal conductivity K. Assuming all changes to be slow, find the distance moved by the piston in time t.
Advertisements
उत्तर
In time dt, heat transfer through the bottom of the cylinder is given by
`"dQ"/"dt" = "KA(T_s - T_0)"/x`
For a monoatomic gas, pressure remains constant.
∴ `dQ = nC_pdT`
∴ `(nC_pdT)/ dt = "KA(T_2 - T_0)"/x`
For a monoatomic gas,
`C_p = 5/2 R`
`⇒ "n5RdT"/"2dt" = KA(T_s - T_0)/x`
`⇒ "5nR"/2 "dT"/dt = (KA(t_s - T_0))/x`
`⇒ "dT"/(T_s - T_0) = "-2KAdt"/"5nRx"`
Integrating both the sides,
`(T_s - T_0)_"T_0"^"T" = "-2KAt"/"5nRx"`
`⇒ In ((T_s - T) /(T_s - T_0)) = - "-2KAt"/"5nRx"`
`⇒ T_s - T = (T_s - T_0)e ^("-2KAt"/"5nRx")`
`⇒ T = T_s - (T_s - T_0) =e ^(-"-2KAt"/"5nRx")`
`⇒ T - T_0 = (T_s - T_0) - (T_s - T_0)e^(-"2KAt"/"5nRx"`
`⇒ T- T_0 = (T_s - _0) [l - e^(-"-2KAt"/"5nRx")]`
From the gas equation,
`(P_(a)Al)/(nR) = T - T_0`
∴ `(P_(a)Al)/(nR)= (T_s - T_0) [1 - e^(-"-2KAt"/"5nRx")]`
`⇒ l = (nR)/(P_aA) (T_s - T_0)[ 1 - e^(-"-2KAt"/"5nRx")]`
APPEARS IN
संबंधित प्रश्न
A bullet of mass 20 g enters into a fixed wooden block with a speed of 40 m s−1 and stops in it. Find the change in internal energy during the process.
The blocks of masses 10 kg and 20 kg moving at speeds of 10 m s−1 and 20 m s−1respectively in opposite directions, approach each other and collide. If the collision is completely inelastic, find the thermal energy developed in the process.
The thermal conductivity of a rod depends on
A hot liquid is kept in a big room. The logarithm of the numerical value of the temperature difference between the liquid and the room is plotted against time. The plot will be very nearly
A piece of charcoal and a piece of shining steel of the same surface area are kept for a long time in an open lawn in bright sun.
(a) The steel will absorb more heat than the charcoal
(b) The temperature of the steel will be higher than that of the charcoal
(c) If both are picked up by bare hand, the steel will be felt hotter than the charcoal
(d) If the two are picked up from the lawn and kept in a cold chamber, the charcoal will lose heat at a faster rate than the steel.
A liquid-nitrogen container is made of a 1 cm thick styrofoam sheet having thermal conductivity 0.025 J s−1 m−1 °C−1. Liquid nitrogen at 80 K is kept in it. A total area of 0.80 m2 is in contact with the liquid nitrogen. The atmospheric temperature us 300 K. Calculate the rate of heat flow from the atmosphere to the liquid nitrogen.
A pitcher with 1-mm thick porous walls contains 10 kg of water. Water comes to its outer surface and evaporates at the rate of 0.1 g s−1. The surface area of the pitcher (one side) = 200 cm2. The room temperature = 42°C, latent heat of vaporization = 2.27 × 106 J kg−1, and the thermal conductivity of the porous walls = 0.80 J s−1 m−1°C−1. Calculate the temperature of water in the pitcher when it attains a constant value.
Water at 50°C is filled in a closed cylindrical vessel of height 10 cm and cross sectional area 10 cm2. The walls of the vessel are adiabatic but the flat parts are made of 1-mm thick aluminium (K = 200 J s−1 m−1°C−1). Assume that the outside temperature is 20°C. The density of water is 100 kg m−3, and the specific heat capacity of water = 4200 J k−1g °C−1. Estimate the time taken for the temperature of fall by 1.0 °C. Make any simplifying assumptions you need but specify them.
Three rods of lengths 20 cm each and area of cross section 1 cm2 are joined to form a triangle ABC. The conductivities of the rods are KAB = 50 J s−1 m−1°C−1, KBC = 200 J s−1m−1°C−1 and KAC = 400 J s−1 m−1°C−1. The junctions A, B and C are maintained at 40°C, 80°C and 80°C respectively. Find the rate of heat flowing through the rods AB, AC and BC.
A hole of radius r1 is made centrally in a uniform circular disc of thickness d and radius r2. The inner surface (a cylinder a length d and radius r1) is maintained at a temperature θ1 and the outer surface (a cylinder of length d and radius r2) is maintained at a temperature θ2 (θ1 > θ2). The thermal conductivity of the material of the disc is K. Calculate the heat flowing per unit time through the disc.
A hollow tube has a length l, inner radius R1 and outer radius R2. The material has a thermal conductivity K. Find the heat flowing through the walls of the tube if (a) the flat ends are maintained at temperature T1 and T2 (T2 > T1) (b) the inside of the tube is maintained at temperature T1 and the outside is maintained at T2.
A composite slab is prepared by pasting two plates of thickness L1 and L2 and thermal conductivites K1 and K2. The slabs have equal cross-sectional area. Find the equivalent conductivity of the composite slab.
Suppose the bent part of the frame of the previous problem has a thermal conductivity of 780 J s−1 m−1 °C−1 whereas it is 390 J s−1 m−1°C−1 for the straight part. Calculate the ratio of the rate of heat flow through the bent part to the rate of heat flow through the straight part.
The two rods shown in following figure have identical geometrical dimensions. They are in contact with two heat baths at temperatures 100°C and 0°C. The temperature of the junction is 70°C. Find the temperature of the junction if the rods are interchanged.
A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at 50°C and 10°C respectively and it is found that 100 J of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.
Two bodies of masses m1 and m2 and specific heat capacities s1 and s2 are connected by a rod of length l, cross-sectional area A, thermal conductivity K and negligible heat capacity. The whole system is thermally insulated. At time t = 0, the temperature of the first body is T1 and the temperature of the second body is T2 (T2 > T1). Find the temperature difference between the two bodies at time t.
