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Question

The ratio of uavg and urms of a gas at a particular temperature is

A
1.1086 : 1
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B
0.92 : 1
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C
1 : 1.1086
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D
1:0.92
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Solution

The correct option is B 0.92 : 1
uavg=8RTπM
urms=3RTM
Since its for the same gas(same molar mass) at same temperature,
uavgurms=83π=0.848=0.92

Theory:

Distribution of molecular speeds :


Plot of 1NdNdu vs u

Area of rectangular strip of width du :

1NdNdu×du=dNN

Where dNN= fraction of molecules in the range of u and u+du

dNN=4π(m2πkT)32emu22kTu2 du

From the plot we can conclude that the number of molecules having very low or very high speed is very less.

Molecular speeds :

Types of molecular speeds :

1.Average Speed (uavg) 2.Most probable speed (ump) 3.Root mean square speed (urms)

Average Speed (uavg):

uavg=u1+u2+........unN

(uavg=28RTπM)

Root mean square speed (urms):

2¯u2=2u21+u22+u23+.........+u2nN

N=Total number of gas molecules

(urms=23RTM)

Most probable speed (ump):

This can be understood with a simple example ,

In the given series 10,9,8,7,6,6,63.6 we see that 6 is the most probable number found in the given series.Similarly speed which is most probable to be acquired by a gas molecule in the given sample is found more frequently .

The speed possessed by the maximum number of molecules at the given temperature.

(ump=22RTM)



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