Advertisements
Advertisements
प्रश्न
Consider a rectangular block of wood moving with a velocity v0 in a gas at temperature T and mass density ρ. Assume the velocity is along x-axis and the area of cross-section of the block perpendicular to v0 is A. Show that the drag force on the block is `4ρAv_0 sqrt((KT)/m)`, where m is the mass of the gas molecule.
Advertisements
उत्तर
Consider the diagram
Let n = number of molecules per unit volume
Vrms = rms speed of the gas molecules
When a block is moving with speed v0, relative speed of molecules w.r.t. front face = v + v0
Coming head-on, momentum transferred to block per collision = 2m (v + v0), where, m = mass of the molecule.
The number of collisions in time Δt = m(v + v0)2nAΔt, where, A = area of cross-section of block and factor of 1/2 appears due to particles moving towards the block.
∴ The momentum transferred in time Δt = m(v + v0)2nAΔt from the front surface.
Similarly, momentum transferred in time Δt = m(v – v0)2nAΔt ......(From the back surface)
∴ Net force (drag force) = mnA[(v + v0)2 – (v – v0)2] .....[From front]
= mnA (4vv0)
= (4 mnAv)v0
= (4ρAv)v0 ......(i)
Where we have assumed ρ = `(mn)/V = M/V`
If v = velocity along the x-axis
Then, we can write KE = `1/2 mv^2 = 1/2 K_BT`
⇒ `v = sqrt((K_BT)/m)` .....`[(K_B = "Boltzmann constant"),(KE = "Kinetic energy"),(T = "Temperature")]`
∴ From equation (i), Drag force = (4ρAv)v0 = `4ρA sqrt((K_BT)/m) v_0`.
APPEARS IN
संबंधित प्रश्न
Do you expect the gas in a cooking gas cylinder to obey the ideal gas equation?
It is said that the assumptions of kinetic theory are good for gases having low densities. Suppose a container is so evacuated that only one molecule is left in it. Which of the assumptions of kinetic theory will not be valid for such a situation? Can we assign a temperature to this gas?
When you come out of a river after a dip, you feel cold. Explain.
The mean square speed of the molecules of a gas at absolute temperature T is proportional to
The temperature and pressure at Simla are 15.0°C and 72.0 cm of mercury and at Kalka these are 35.0°C and 76.0 cm of mercury. Find the ratio of air density at Kalka to the air density at Simla.
Use R=8.314J K-1 mol-1
0.040 g of He is kept in a closed container initially at 100.0°C. The container is now heated. Neglecting the expansion of the container, calculate the temperature at which the internal energy is increased by 12 J.
Use R = 8.3 J K-1 mol-1
An ideal gas is trapped between a mercury column and the closed-end of a narrow vertical tube of uniform base containing the column. The upper end of the tube is open to the atmosphere. The atmospheric pressure equals 76 cm of mercury. The lengths of the mercury column and the trapped air column are 20 cm and 43 cm respectively. What will be the length of the air column when the tube is tilted slowly in a vertical plane through an angle of 60°? Assume the temperature to remain constant.
Figure shows two rigid vessels A and B, each of volume 200 cm3, containing an ideal gas (Cv = 12.5 J K−1 mol−1). The vessels are connected to a manometer tube containing mercury. The pressure in both the vessels is 75 cm of mercury and the temperature is 300 K. (a) Find the number of moles of the gas in each vessel. (b) 5.0 J of heat is supplied to the gas in vessel A and 10 J to the gas in vessel B. Assuming there's no appreciable transfer of heat from A to B, calculate the difference in the heights of mercury in the two sides of the manometer. Gas constant, R = 8.3 J K−1 mol−1.
Answer in brief:
Show that rms velocity of an oxygen molecule is `sqrt2` times that of a sulfur dioxide molecule at S.T.P.
When a gas is heated, its temperature increases. Explain this phenomenon on the basis of the kinetic theory of gases.
Calculate the ratio of the mean square speeds of molecules of a gas at 30 K and 120 K.
The number of degrees of freedom, for the vibrational motion of a polyatomic molecule, depends on the ______
Under which condition laws of Boyle, Charles, and Gay-Lussac are valid?
Average kinetic energy of H2 molecule at 300K is 'E'. At the same temperature, average kinetic energy of O2 molecule will be ______.
A cylinder containing an ideal gas is in vertical position and has a piston of mass M that is able to move up or down without friction (Figure). If the temperature is increased ______.
Two molecules of a gas have speeds of 9 × 10 6 ms−1 and 1 × 106 ms−1, respectively. What is the root mean square speed of these molecules?
A gas mixture consists of molecules of types A, B and C with masses mA > mB > mC. Rank the three types of molecules in decreasing order of average K.E.
An insulated container containing monoatomic gas of molar mass m is moving with a velocity vo. If the container is suddenly stopped, find the change in temperature.
Which of the following materials is diathermanous?
