<< Chapter < Page Chapter >> Page >

Calculating temperature: calorimetry with an ideal gas

A 300-g piece of solid gallium (a metal used in semiconductor devices) at its melting point of only 30.0 ° C is in contact with 12.0 moles of air (assumed diatomic) at 95.0 ° C in an insulated container. When the air reaches equilibrium with the gallium, 202 g of the gallium have melted. Based on those data, what is the heat of fusion of gallium? Assume the volume of the air does not change and there are no other heat transfers.

Strategy

We’ll use the equation Q hot + Q cold = 0 . As some of the gallium doesn’t melt, we know the final temperature is still the melting point. Then the only Q hot is the heat lost as the air cools, Q hot = n air C V Δ T , where C V = 5 R / 2 . The only Q cold is the latent heat of fusion of the gallium, Q cold = m Ga L f . It is positive because heat flows into the gallium.

Solution

  1. Set up the equation:
    n air C V Δ T + m Ga L f = 0 .
  2. Substitute the known values and solve:
    ( 12.0 mol ) ( 5 2 ) ( 8.31 J mol · ° C ) ( 30.0 ° C 95.0 °C ) + ( 0.202 kg ) L f = 0 .

    We solve to find that the heat of fusion of gallium is 80.2 kJ/kg.
Got questions? Get instant answers now!

Summary

  • Every degree of freedom of an ideal gas contributes 1 2 k B T per atom or molecule to its changes in internal energy.
  • Every degree of freedom contributes 1 2 R to its molar heat capacity at constant volume C V .
  • Degrees of freedom do not contribute if the temperature is too low to excite the minimum energy of the degree of freedom as given by quantum mechanics. Therefore, at ordinary temperatures, d = 3 for monatomic gases, d = 5 for diatomic gases, and d 6 for polyatomic gases.

Conceptual questions

Experimentally it appears that many polyatomic molecules’ vibrational degrees of freedom can contribute to some extent to their energy at room temperature. Would you expect that fact to increase or decrease their heat capacity from the value R ? Explain.

Got questions? Get instant answers now!

One might think that the internal energy of diatomic gases is given by E int = 5 R T / 2 . Do diatomic gases near room temperature have more or less internal energy than that? Hint: Their internal energy includes the total energy added in raising the temperature from the boiling point (very low) to room temperature.

Less, because at lower temperatures their heat capacity was only 3 RT /2.

Got questions? Get instant answers now!

You mix 5 moles of H 2 at 300 K with 5 moles of He at 360 K in a perfectly insulated calorimeter. Is the final temperature higher or lower than 330 K?

Got questions? Get instant answers now!

Problems

To give a helium atom nonzero angular momentum requires about 21.2 eV of energy (that is, 21.2 eV is the difference between the energies of the lowest-energy or ground state and the lowest-energy state with angular momentum). The electron-volt or eV is defined as 1.60 × 10 −19 J . Find the temperature T where this amount of energy equals k B T / 2 . Does this explain why we can ignore the rotational energy of helium for most purposes? (The results for other monatomic gases, and for diatomic gases rotating around the axis connecting the two atoms, have comparable orders of magnitude.)

4.92 × 10 5 K ; Yes, that’s an impractically high temperature.

Got questions? Get instant answers now!

(a) How much heat must be added to raise the temperature of 1.5 mol of air from 25.0 °C to 33.0 °C at constant volume? Assume air is completely diatomic. (b) Repeat the problem for the same number of moles of xenon, Xe.

Got questions? Get instant answers now!

A sealed, rigid container of 0.560 mol of an unknown ideal gas at a temperature of 30.0 °C is cooled to −40.0 °C . In the process, 980 J of heat are removed from the gas. Is the gas monatomic, diatomic, or polyatomic?

polyatomic

Got questions? Get instant answers now!

A sample of neon gas (Ne, molar mass M = 20.2 g/mol ) at a temperature of 13.0 °C is put into a steel container of mass 47.2 g that’s at a temperature of −40.0 °C . The final temperature is −28.0 °C . (No heat is exchanged with the surroundings, and you can neglect any change in the volume of the container.) What is the mass of the sample of neon?

Got questions? Get instant answers now!

A steel container of mass 135 g contains 24.0 g of ammonia, NH 3 , which has a molar mass of 17.0 g/mol. The container and gas are in equilibrium at 12.0 °C . How much heat has to be removed to reach a temperature of −20.0 °C ? Ignore the change in volume of the steel.

3.08 × 10 3 J

Got questions? Get instant answers now!

A sealed room has a volume of 24 m 3 . It’s filled with air, which may be assumed to be diatomic, at a temperature of 24 °C and a pressure of 9.83 × 10 4 Pa . A 1.00-kg block of ice at its melting point is placed in the room. Assume the walls of the room transfer no heat. What is the equilibrium temperature?

Got questions? Get instant answers now!

Heliox, a mixture of helium and oxygen, is sometimes given to hospital patients who have trouble breathing, because the low mass of helium makes it easier to breathe than air. Suppose helium at 25 °C is mixed with oxygen at 35 °C to make a mixture that is 70 % helium by mole. What is the final temperature? Ignore any heat flow to or from the surroundings, and assume the final volume is the sum of the initial volumes.

29.2 °C

Got questions? Get instant answers now!

Professional divers sometimes use heliox, consisting of 79 % helium and 21 % oxygen by mole. Suppose a perfectly rigid scuba tank with a volume of 11 L contains heliox at an absolute pressure of 2.1 × 10 7 Pa at a temperature of 31 °C . (a) How many moles of helium and how many moles of oxygen are in the tank? (b) The diver goes down to a point where the sea temperature is 27 °C while using a negligible amount of the mixture. As the gas in the tank reaches this new temperature, how much heat is removed from it?

Got questions? Get instant answers now!

In car racing, one advantage of mixing liquid nitrous oxide ( N 2 O ) with air is that the boiling of the “nitrous” absorbs latent heat of vaporization and thus cools the air and ultimately the fuel-air mixture, allowing more fuel-air mixture to go into each cylinder. As a very rough look at this process, suppose 1.0 mol of nitrous oxide gas at its boiling point, −88 °C , is mixed with 4.0 mol of air (assumed diatomic) at 30 °C . What is the final temperature of the mixture? Use the measured heat capacity of N 2 O at 25 °C , which is 30.4 J/mol °C . (The primary advantage of nitrous oxide is that it consists of 1/3 oxygen, which is more than air contains, so it supplies more oxygen to burn the fuel. Another advantage is that its decomposition into nitrogen and oxygen releases energy in the cylinder.)

−1.6 °C

Got questions? Get instant answers now!

Questions & Answers

what is defense mechanism
Chinaza Reply
what is defense mechanisms
Chinaza
I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 2' conversation and receive update notifications?

Ask