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Question

# For gaseous state, if the most probable speed is denoted by C∗, average speed by ¯C and root mean square speed by C, then for a large number of molecules, the ratios of these speeds are:

A
C¯CC1.2251.1281
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B
C¯CC11.1281.225
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C
C¯CC1.1281.2251
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D
C¯CC11.1251.128
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Solution

## The correct option is B C∗¯CC11.1281.225Using thr formula, C∗=√2RTM; ¯C=√8RTπM; C=√3RTM C∗:¯C:C=√2:√8π:√3=1.414:1.596:1.732 =1:1.128:1.225 Theory: Kinetic gas equation : PV=13mN¯u2 Volume of container(V)=l3 Verification of gas laws with kinetic gas equation : PV=13mN¯u2.......................(1) According to Kinetic theory of gases : 12m¯u2rms=λT (in kelvin).................(2) where λ is a proportionality constant. Multiply and divide equation 1 by 2 on both sides : PV=2312mN¯u2................(3) From equation (2) 12m¯u2rms=λT.......(4) Putting the value of equation 4 in equation 3 : PV=23(λT)N P∝1V which is Boyle's law at constant N and T. Similarly we can prove charles’ law from kinetic gas equation : At constant N and Pressure(P): V∝T and similarly it can be done at constant N and Volume (V) which is Gay lussac’s law : P∝T At constant temperature (T) and Pressure (P) Kinetic gas equation can prove Avogadro’s law : V∝N or V=k3N Dividing and multiplying by NA V=k3NA(NNA) or V=k4(NNA) Calculation of λ : From ideal gases : PV=nRT=23λNT........(1) Where n=number of moles of gas andNA=Avogadro′s number Since number of moles(n)=Given number of molecules(N)Avogadro′s number(NA) N=nNA...........(2) Putting the value of N from 2 in 1 : PV=nRT=23(nNA)λT λ=32RNA or λ=32k where k is Boltzmann constant k(Boltzmann constant)=RNA Average kinetic energy of molecules : 12m¯u2rms∝T(in kelvin)or 12m¯u2rms=λT(in kelvin) where λ is a proportionality constant. λT=32kT where k is Boltzmann constant k=1.38×10−23 JK−1 Total Kinetic energy per mole : Total kinetic energy=NA(12m¯u2rms)=NA(λT) =32(RNA)NA=32R Root mean square speed Vrms : Total kinetic energy=12M¯u2rms=32RT ¯u2rms=3RTM or ¯urms=2√3RTM Unit of urms: Unit of urms=√(J K−1 mol−1)(K)kg mol−1 Unit of urms=√Jkg Unit of urms=√kg m2 s−2kg Unit of urms=m s−1 Note: Always take R in J K−1 mol−1 and M in kg mol−1

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