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Chapter 19: Q30 (page 579)

Question: The mean free path of nitrogen molecules at 0.0⁰C and 1atmis $0.80 脳 10^{- 5} c m$ . At this temperature and pressure there are $2.7 脳 10^{19} m o l e c u l e s / c m^{3}$ . What is the molecular diameter?

Short Answer

Expert verified

Answer

The molecular diameter of nitrogen molecule is . 0.32 nm

Step by step solution

01

Given data

Atmospheric density

$\frac{N}{V} = 2.7 脳 10^{19} \frac{molecule}{{cm}^{3}} = 2.7 脳 10^{25} \frac{molecule}{m^{3}}$

Temperature;

$T = 0.0 掳 C$

Mean free path

$位 = 0.80 脳 10^{- 5} cm = 0.80 脳 10^{- 7} m$

Pressure

$P = 1.0 atm$

02

Understanding the concept

The expression for the mean free path is given by,

$位 = \frac{1}{\sqrt{2} 蟺 d^{2} (\frac{N}{V})}$

Here $位$ is the mean free path, $\frac{N}{V}$ is the molecule density in $m o l e c u l e / c m^{3}$ .

03

Calculate the molecular diameter

The mean free path is given by

$位 = \frac{1}{\sqrt{2} 蟺 d^{2} (\frac{N}{V})}$

On rearranging the equation in terms of d

$d = \sqrt{\frac{1}{\sqrt{2} 蟺位 (\frac{N}{V})}}$

Substitute $0.80 脳 10^{- 7} 尘蹿辞谤位$ and $2.7 脳 10^{25} \frac{molecule}{m^{3}} f o r \frac{位}{v}$ into the above equation

$d = \sqrt{\frac{1}{\sqrt{2} (3.142) (0.80 脳 10^{- 7}) (2.7 脳 10^{25})}} d = 3.22 脳 10^{- 10} m = 0.32 nm$

Therefore the molecular diameter of nitrogen molecule is 0.32 nm .

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Most popular questions from this chapter

Suppose $1.80鈥尘辞濒$ of an ideal gas is taken from a volume of ${3.00鈥尘}^{3}$ to a volume of $1.5 鈥� m^{3}$ via an isothermal compression at $30 掳 C$

How much energy is transferred as heat during the compression
Is the transfer to or from the gas?

Figure 19-28 shows a hypothetical speed distribution for particles of a certain gas: P(v)= Cv² for $0 < v 鈮� v_{0}$ and P(v) = 0 for $v > v_{0}$ . Find

(a) an expression for Cin terms of v₀,

(b) the average speed of the particles, and

(c) their rms speed.

A sample of ideal gas expands from an initial pressure and volume of 32atmand $1.0 L$ to a final volume of $4.0 鈥� L$ . The initial temperature is $300 鈥� K$ . If the gas is monatomic and the expansion isothermal, what are the (a) final pressure $P_{f}$ , (b) final temperature $T_{f}$ , and (c) work W done by the gas? If the gas is monatomic and the expansion adiabatic, what are (d) $P_{f}$ , (e) $T_{f}$ , and (f) W? If the gas is diatomic and the expansion adiabatic, what are (g) $P_{f}$ , (h) $T_{f}$ , and (i) W?

At $20 掳 C$ and $750 t o r r$ pressure, the mean free paths for argon gas (Ar) and nitrogen gas ( $N_{2}$ ) ARE $位_{A r} = 9.9 脳 10^{- 6} c m$ and $位_{N 2} = 27.5 脳 10^{- 6} 鈥� c m$ .

a) Find the ratio of the diameter of $A_{r}$ an atom to that of an $N_{2}$ molecule.

b) What is the mean free path of argon at $20 掳 C$ and $150 t o r r$ ?

c) What is the mean free path of argon at $- 40 掳 C$ and $750 t o r r$ ?

Suppose 12.0gof oxygen (O₂) gas is heated at constant atmospheric pressure from $25 . 0^{鈭�} C to 125 . 0^{鈭�} C$ to.

How many moles of oxygen are present?
How much energy is transferred to the oxygen as heat? (The molecules rotate but do not oscillate).
What fraction of heat is used to raise the internal energy of the oxygen?

See all solutions

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