Advertisements
Advertisements
Question
Following figure shows two adiabatic vessels, each containing a mass m of water at different temperatures. The ends of a metal rod of length L, area of cross section A and thermal conductivity K, are inserted in the water as shown in the figure. Find the time taken for the difference between the temperatures in the vessels to become half of the original value. The specific heat capacity of water is s. Neglect the heat capacity of the rod and the container and any loss of heat to the atmosphere.
Advertisements
Solution
Rate of transfer of heat from the rod is given as
`(DeltaQ)/(Deltat) = (KA(T_1 - T_2))/l`
In time t, the temperature difference becomes half.
In time Δt, the heat transfer from the rod will be given by
`DeltaQ = (KA(T_1 - T_2))/l Deltat`............(i)
Heat loss by water at temperature T1 is equal to the heat gain by water at temperatureT2.
Therefore, heat loss by water at temperature T1 in time `Deltat` is given by
`DeltaQ=ms(T_1 - T_1) ............(ii)`
From equation (i) and (ii),
ms `( T_1 - T_1' ) = (KA(T_1 - T_2))/l Delta t`
`⇒ T_1' = T_1 - (KA(T_1 - T_2))/(l(ms) Delta t`
This gives us the fall in the temperature of water at temperature T1.
Similarly, rise in temperature of water at temperature T2 is given by
`T_2'= T_2 +( KA(T_1 - T_2))/(l(ms) )Delta t`
Change in the temperature is given by
`(T_1' - T_2' ) = ( T_1 - T_2)- 2 (KA(T_1 - T_2))/(l(ms)) Deltat`
`⇒ (T_1 - T_2) - ( T_1' - T_2') = 2 (KA(T_1 - T_2))/(l(ms)) Deltat`
`⇒ (DeltaT)/(Deltat) = 2 (KA(T_1 - T_2))/(l(ms)`
Here, `(DeltaT)/(Deltat)` is the rate of change of temperature difference.
Taking limit Δ t→ 0,
`⇒ (dT)/dt = 2 (KA (T_1 - T_2))/ (l(ms))`
`rArr (dT)/(T_1 - T_2 )= 2(KA)/(l(ms)dt`
On integrating within proper limit, we get
\[\int_{(T_1 - T_2)}^{(T_1 - T_2)/2}\] `(dT)/((T_1 - T_2)) = 2 (KA)/(l(ms))` \[\int\limits_0^t dt\]
`⇒ In [((T_1- T_2))/(2(T_1 - t_2)]] = 2 = (KA)/(l(ms))t `
`⇒In [ 2] = 2(KA)/(l(ms))t`
`⇒ t = In [2] (lms)/(2KA)`
APPEARS IN
RELATED QUESTIONS
Explain why a brass tumbler feels much colder than a wooden tray on a chilly day
A tightly closed metal lid of a glass bottle can be opened more easily if it is put in hot water for some time. Explain.
A metal sheet with a circular hole is heated. The hole
Find the ratio of the lengths of an iron rod and an aluminium rod for which the difference in the lengths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 12 × 10–6 °C–1 and 23 × 10–6 °C–1 respectively.
A glass window is to be fit in an aluminium frame. The temperature on the working day is 40°C and the glass window measures exactly 20 cm × 30 cm. What should be the size of the aluminium frame so that there is no stress on the glass in winter even if the temperature drops to 0°C? Coefficients of linear expansion for glass and aluminium are 9.0 × 10–6 °C–1 and 24 ×100–6°C–1 , respectively.
In a room containing air, heat can go from one place to another
One end of a steel rod (K = 46 J s−1 m−1°C−1) of length 1.0 m is kept in ice at 0°C and the other end is kept in boiling water at 100°C. The area of cross section of the rod is 0.04 cm2. Assuming no heat loss to the atmosphere, find the mass of the ice melting per second. Latent heat of fusion of ice = 3.36 × 105 J kg−1.
Following Figure shows water in a container having 2.0 mm thick walls made of a material of thermal conductivity 0.50 W m−1°C−1. The container is kept in a melting-ice bath at 0°C. The total surface area in contact with water is 0.05 m2. A wheel is clamped inside the water and is coupled to a block of mass M as shown in the figure. As the block goes down, the wheel rotates. It is found that after some time a steady state is reached in which the block goes down with a constant speed of 10 cm s−1 and the temperature of the water remains constant at 1.0°C. Find the mass M of the block. Assume that the heat flows out of the water only through the walls in contact. Take g = 10 m s−2.
On a winter day when the atmospheric temperature drops to −10°C, ice forms on the surface of a lake. (a) Calculate the rate of increase of thickness of the ice when 10 cm of the ice is already formed. (b) Calculate the total time taken in forming 10 cm of ice. Assume that the temperature of the entire water reaches 0°C before the ice starts forming. Density of water = 1000 kg m−3, latent heat of fusion of ice = 3.36 × 105 J kg−1and thermal conductivity of ice = 1.7 W m−1°C−1. Neglect the expansion of water of freezing.
Steam at 120°C is continuously passed through a 50 cm long rubber tube of inner and outer radii 1.0 cm and 1.2 cm. The room temperature is 30°C. Calculate the rate of heat flow through the walls of the tube. Thermal conductivity of rubber = 0.15 J s−1 m−1°C−1.
Four identical rods AB, CD, CF and DE are joined as shown in following figure . The length, cross-sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperature T1, T2 and T3 respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.
A calorimeter of negligible heat capacity contains 100 cc of water at 40°C. The water cools to 35°C in 5 minutes. The water is now replaced by K-oil of equal volume at 40°C. Find the time taken for the temperature to become 35°C under similar conditions. Specific heat capacities of water and K-oil are 4200 J kg−1 K−1 and 2100 J kg−1 K−1respectively. Density of K-oil = 800 kg m−3.
There are two identical vessels filled with equal amounts of ice. The vessels are of different metals. If the ice melts in the two vessels in 20 and 35 minutes respectively, the ratio of the coefficients of thermal conductivity of the two metals is:
The coefficient of thermal conductivity depends upon ______.
These days people use steel utensils with copper bottom. This is supposed to be good for uniform heating of food. Explain this effect using the fact that copper is the better conductor.
In conduction, how does heat mainly travel through a solid rod kept with one end in a flame?
Why are metals like copper and aluminium called good conductors of heat?
During conduction along a metal rod, which part becomes hot first, and why?
