Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
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.
Two identical rectangular strips, one of copper and the other of steel, are riveted together to form a bimetallic strip (acopper> asteel). On heating, this strip will
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.
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.
A semicircular rod is joined at its end to a straight rod of the same material and the same cross-sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Find the ratio of the heat transferred through a cross section of the semicircular rod to the heat transferred through a cross section of the straight rod in a given time.
A room has a window fitted with a single 1.0 m × 2.0 m glass of thickness 2 mm. (a) Calculate the rate of heat flow through the closed window when the temperature inside the room is 32°C and the outside is 40°C. (b) The glass is now replaced by two glasspanes, each having a thickness of 1 mm and separated by a distance of 1 mm. Calculate the rate of heat flow under the same conditions of temperature. Thermal conductivity of window glass = 1.0 J s−1 m−1°C−1 and that of air = 0.025 m-1°C-1 .
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.
A thin rod having length L0 at 0°C and coefficient of linear expansion α has its two ends maintained at temperatures θ1 and θ2, respectively. Find its new length.
According to Stefan’s law of radiation, a black body radiates energy σT4 from its unit surface area every second where T is the surface temperature of the black body and σ = 5.67 × 10–8 W/m2K4 is known as Stefan’s constant. A nuclear weapon may be thought of as a ball of radius 0.5 m. When detonated, it reaches temperature of 106 K and can be treated as a black body.
- Estimate the power it radiates.
- If surrounding has water at 30°C, how much water can 10% of the energy produced evaporate in 1s? [Sw = 4186.0 J/kg K and Lv = 22.6 × 105 J/kg]
- If all this energy U is in the form of radiation, corresponding momentum is p = U/c. How much momentum per unit time does it impart on unit area at a distance of 1 km?
Which of the following is a correct example of a bad conductor of heat used in daily life to reduce heat loss or gain?
During conduction along a metal rod, which part becomes hot first, and why?
