1. A sphere of radius R contains water. If you make a small hole of radius r at

Question

1. A sphere of radius R contains water. If you make a small hole of radius r at the bottom derive the time that it would take to empty the sphere. If R = 1 m and r = 0.01 m, calculate the time. 2. Use the conservation of momentum gas molecule. Find the rms velocity of a nitrogen molecule if the room temperature is 27 degree principle to calculate the individual kinetic energy of a Celsius. 3. Derive the mean free time and mean free path of a gas molecule. Calculate both these parameters for a nitrogen molecule at T 27 degree Celsius. 4. Derive the expression for the rms velocity using the Maxwell-Boltzmann distribution. 5. Prove that the integral of M-B distribution from zero to infinity is 1. What will be the fraction of nitrogen molecules at T = 300 K that will have velocities between 100 m/s and 200 m/s? ..You have decided to freeze water in your freezer. There are 16 small containers in a 10 cm x 20 cm tray with a depth of 3 cm. If the freezer temperature is -10 degree Celsius and the water temperature- initially - was 20 degree Celsius, find the time it will take to freeze the water completely. Assume the heat escapes only through the top of the water surface. A steel pendulum clock of length 1 m gives correct time at T-20 Celsius. Find the maximum or minimum) temperature that can be allowed so that the clock does not run more than 4 second fast or slow in a day. Assume when an air molecule strikes your body it gives up all its energy. Calculate the number of air molecules needed to raise body mass to be 50 kg and consisting of water your body temperature by 1 degree. Assume your

Explanation / Answer

let say there is a container of water of dimensions a x b and depth d which is left to freeze at outside temperature T and inside temperature of 0 C
then after time t depth y has frozen
at this time
conductivity of ice k = 2.3
so heat lost through the ice in time dt
dq = k*a*b*(-T)*dt/y [ where T < 0)]

also, this heat freezes dy thickness of layer of water
latent heat of fusioon of water L: = 334,000 J / kg
so, dq = L*dm
but dm = rho*a*b*dy
rho = density of ice = 916.7 kg / m^3

so, rho*a*b*dy*L = k*a*b(-T)*dt/y
rho*y*dy*L = k*(-T)*dt

integrating from y = 0 to y = d, t = 0 to t = t
rho*d^2*L/2 = k(-T)t

from the given data
T = -10 C
d = 0.03 m
so, 916.7*(0.03^2/2)*334000 = 2.3*10*t
t = 5990.435 s = 93.173 min( time taken to freeze water at 0 C)

also, time taken for water to reach 0 deg from 20 C = T
then fron newtons law of cooling
temperature at time t is
T(t) = To + (Ti - To)e^(-kt)
where Ti is inside temperature = 20 C
To is outside temp = -10 C
T(t) = 0 C
k = 0.0324 per minute for water
so, 0 = -10 + (20 + 10)e^(-0.0324t)
e^(-0.0324t) = 10/30
t = 33.9077 min

so total time to freezw = 93.173 + 33.9077 = 127.0816 min

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