Specific Heat Capacities of Gases
EXERCISES, Q31-Q35
Taking log to both sides,
EXERCISES, Q31-Q35
31. An adiabatic cylinder tube of cross-sectional area 1 cm² is closed at one end and fitted with a piston at the other end. The tube contains 0.03 g of an ideal gas. At 1 atm pressure and at the temperature of the surrounding, the length of the gas column is 40 cm. The piston is suddenly pulled out to double the length of the column. The pressure of the gas falls to 0.355 atm. Find the speed of sound in the gas at atmospheric temperature.
Answer: (a) m = 0.03 g =3x10⁻⁵ kg,
Area of the piston, A = 1 cm² =1x10⁻⁴ m²,
Length, L = 40 cm =0.40 m.
Initial volume, V =AL
=1x10⁻⁴*0.40 =4x10⁻⁵ m³.
Initial density, ρ =m/V
=3x10⁻⁵/4x10⁻⁵ kg/m³
=0.75 kg/m³
Initial pressure, p =1atm =1x10⁵ Pa.
Final volume, V' = 2V
Final pressure, p' = 0.355 atm
If Cₚ/Cᵥ =ˠ for the gas, the in the above adiabatic process,
p'V'ˠ = pVˠ
→(2V)ˠ = (p/p')Vˠ
→2ˠ = (1/0.355) = 2.817
Taking log to both sides,
ˠln2 =ln2.817
→ˠ =ln2.817/ln2 = 1.5
The speed of sound in a gas is given as,
v = √(ˠp/ρ)
=√{1.5*1x10⁵/0.75}
=√2x10⁵
=447 m/s.
32. The speed of sound in hydrogen at 0°C is 1280 m/s. The density of hydrogen at STP is 0.089 kg/ m³. Calculate the molar heat capacities Cₚ and Cᵥ of hydrogen.
Answer: The density of hydrogen at STP,
ρ =0.089 kg/m³.
Speed of sound at this temperature,
v =1280 m/s, but,
v =√(ˠp/ρ)
→ˠ =ρv²/p =0.089*1280²/10⁵ =1.46
Since R = 8.3 J/mol-K,
Cᵥ = R/(ˠ-1) =8.3/0.46
= 18.0 J/mol-K.
And Cₚ = ˠCᵥ =1.46*18.0
= 26.3 J/mol-K.
33. 4.0 g of helium occupies 22400 cm³ at STP. The specific heat capacity of helium at constant pressure is 5.0 cal/mol-K. Calculate the speed of sound in helium at STP.
Answer: Density of helium at STP,
ρ = 4*10⁶/22400*1000
= 0.179 kg/m³
Cₚ =5.0 cal/mol-K
=5.0*4.2 J/mol-K
= 21.0 J/mol-K.
Since Cₚ =ˠR/(ˠ-1)
→21 = ˠ*8.3/(ˠ-1)
→21ˠ -21 = 8.3ˠ
→12.7ˠ =21
→ˠ = 21/12.7 =1.65
At STP, the pressure of helium,
p = 10⁵ Pa.
Speed of sound in helium at STP,
v = √(ˠp/ρ)
=√(1.65*10⁵/0.179)
=960 m/s.
34. An ideal gas having density 1.7x10⁻³ g/cm³ at a pressure 1.5x10⁵ Pa is filled in a Kundt's tube. When the gas is resonated at a frequency of 3.0 kHz, nodes are formed at a separation of 6.0 cm. Calculate the molar heat capacities Cₚ and Cᵥ of the gas.
Answer: The resonating frequency,
f = 3.0 kHz =3000 Hz.
Pressure, p = 1.5x10⁵ Pa
Since the nodes are formed at a separation of 6.0 cm, it means,
λ/2 = 6.0 cm = 0.06 m, where λ is the wavelength of sound.
→λ = 2*0.06 =0.12 m.
Hence speed of sound, v =fλ
→v = 3000*0.12 m/s
= 360 m/s.
Given ρ =1.7x10⁻³ g/cm³
=1.7x10⁻³*10⁶/1000 kg/m³
=1.7 kg/m³
Now the speed of sound in gas is given as, v =√(ˠp/ρ)
→ˠ = ρv²/p
= 1.7*360²/1.5x10⁵
= 1.47
Now Cᵥ = R/(ˠ-1) =8.3/0.47
=17.7 J/mol-K.
And Cₚ =ˠ*Cᵥ = 1.47*17.7
=26.0 J/mol-K.
35. Standing waves of frequency 5.0 kHz are produced in a tube filled with oxygen at 300 K. The separation between the consecutive nodes is 3.3 cm. Calculate the specific heat capacities Cₚ and Cᵥ of the gas.
Answer: The frequency of sound waves, f = 5.0 kHz = 5000 Hz.
The separation between the consecutive nodes = 3.3 cm = 3.3x10⁻² m. If λ is the wavelength of the sound waves, then
λ/2 = 3.3x10⁻² m
→λ = 6.6x10⁻² m.
Hence the speed of sound in the given sample of oxygen, v = fλ.
→v =5000*6.6x10⁻² m/s
= 330 m/s.
Given T = 300 K.
Now, ρ = m/V
Since v =√(ˠp/ρ)
→v² = ˠ*(nRT/V)/(m/V)
= ˠRT(n/m)
= ˠRT(m/Mm)
= ˠRT/M
→ˠ = Mv²/RT
{For oxygen, M = 32 g =32x10⁻³ kg}
= 32x10⁻³*330²/(8.3*300)
= 1.4
Now Cₚ = ˠR/(ˠ-1)
= 1.4*8.3/(1.4-1)
= 29.0 J/mol-K.
And Cᵥ = Cₚ/ˠ =29.0/1.4
= 20.7 J/mol-K.
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CHAPTER- 27-Specific Heat Capacities of Gases
CHAPTER- 26-Laws of Thermodynamics
CHAPTER- 25-CALORIMETRY
Questions for Short Answer
OBJECTIVE-I
OBJECTIVE-II
EXERCISES - Q-11 to Q-18
CHAPTER- 24-Kinetic Theory of Gases
CHAPTER- 23 - Heat and Temperature
CHAPTER- 21 - Speed of Light
CHAPTER- 20 - Dispersion and Spectra
CHAPTER- 19 - Optical Instruments
CHAPTER- 18 - Geometrical Optics
CHAPTER- 17 - Light Waves
CHAPTER- 16 - Sound Waves
CHAPTER- 15 - Wave Motion and Waves on a String
CHAPTER- 14 - Fluid Mechanics
CHAPTER- 13 - Fluid Mechanics
CHAPTER- 12 - Simple Harmonic Motion
CHAPTER- 11 - Gravitation
CHAPTER- 10 - Rotational Mechanics
CHAPTER- 9 - Center of Mass, Linear Momentum, Collision
CHAPTER- 27-Specific Heat Capacities of Gases
CHAPTER- 26-Laws of Thermodynamics
CHAPTER- 25-CALORIMETRY
Questions for Short Answer
OBJECTIVE-I
OBJECTIVE-II
CHAPTER- 24-Kinetic Theory of Gases
CHAPTER- 23 - Heat and Temperature
CHAPTER- 21 - Speed of Light
CHAPTER- 20 - Dispersion and Spectra
CHAPTER- 19 - Optical Instruments
CHAPTER- 18 - Geometrical Optics
CHAPTER- 17 - Light Waves
CHAPTER- 16 - Sound Waves
CHAPTER- 15 - Wave Motion and Waves on a String
CHAPTER- 14 - Fluid Mechanics
CHAPTER- 13 - Fluid Mechanics
CHAPTER- 12 - Simple Harmonic Motion
CHAPTER- 11 - Gravitation
CHAPTER- 10 - Rotational Mechanics
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CHAPTER- 2 - "Physics and Mathematics"
CHAPTER- 2 - "Physics and Mathematics"
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