General Physics (PHY 2130)

Transcription

1 General Physics (PHY 2130) Lecture 33 Heat Ideal gases Phase transitions. Latent heat.

2 Lightning Review Last lecture: 1. Thermal physics Heat. Specific heat. Review Problem: Doppler ultrasound is used to measure the speed of blood flow. If the speed of the red blood cells is v, the speed of sound in blood is u, the ultrasound source emits waves of frequency f, and we assume that the blood cells are moving directly toward the ultrasound source, then the frequency f r of reflected waves detected by the apparatus is given by the formula below. The reflected sound interferes with the emitted sound, producing beats. If the speed of red blood cells is 0.11 m/s, the ultrasound frequency used is 4.6 MHz, and the speed of sound in blood is 1570 m/s, what is the beat frequency?

3 3 Two waves of different frequency Superposition of the above waves The beat frequency is Δf f 1 f 2

4 Doppler ultrasound is used to measure the speed of blood flow. If the speed of the red blood cells is v, the speed of sound in blood is u, the ultrasound source emits waves of frequency f, and we assume that the blood cells are moving directly toward the ultrasound source, then the frequency f r of reflected waves detected by the apparatus is given by the formula below. 4 The reflected sound interferes with the emitted sound, producing beats. If the speed of red blood cells is 0.11 m/s, the ultrasound frequency used is 4.6 MHz, and the speed of sound in blood is 1570 m/s, what is the beat frequency? The beat frequency is given by the difference of frequencies of emitted and reflected ultrasound waves:

5 Specific Heat Every substance requires a unique amount of energy per unit mass to change the temperature of that substance by 1 C directly proportional to mass (thus, per unit mass) The specific heat, c, of a substance is a measure of this amount c C m Q m ΔT Units SI Joule/kg C (J/kg C) CGS Calorie/g C (cal/g C )

6 Notes: Heat and Specific Heat Q m c ΔT ΔT is always the final temperature minus the initial temperature When the temperature increases, ΔT and ΔQ are considered to be positive and energy flows into the system When the temperature decreases, ΔT and ΔQ are considered to be negative and energy flows out of the system

7 Example: A kg aluminum teakettle contains 2.00 kg of water at 15.0 C. How much heat is required to raise the temperature of the water (and kettle) to 100 C? 7 The heat needed to raise the temperature of the water to T f is Q ( 2 kg)( kj/kg C)( 85 C) 712 kj. w mwcwδtw The heat needed to raise the temperature of the aluminum to T f is Q ( 0.4 kg)( kj/kg C)( 85 C) 30.6 kj. Al malcalδtal Then Q total Q w + Q Al 732 kj.

8 Consequences of Different Specific Heats Water has a high specific heat compared to land On a hot day, the air above the land warms faster The warmer air flows upward and cooler air moves toward the beach What happens at night? c c Si H O 700 J J kg kg C C

9 Question What happens at night? 1. same 2. opposite 3. nothing 4. none of the above How to determine specific heat?

10 Calorimeter A technique for determining the specific heat of a substance is called calorimetry A calorimeter is a vessel that is a good insulator that allows a thermal equilibrium to be achieved between substances without any energy loss to the environment

11 Calorimetry Analysis performed using a calorimeter Conservation of energy applies to the isolated system The energy that leaves the warmer substance equals the energy that enters the water Q cold -Q hot Negative sign keeps consistency in the sign convention of ΔT

12 Example: A kg piece of unknown metal heated to 100 C and dropped into the bucket containing 0.5 kg of water at 20 C. Determine specific heat of metal if the final temperature of the system is 50 C Given: Mass: m kg m kg Specific heat (water): c W 4186 J/kg C Temperatures: T C T 2 20 C T f 50 C Find: Specific heat? Q water Conservation of energy: heat lost by metal is the same as heat acquired by water: + Q Solve this equation: metal 0 mmetalcmetalδtmetal + mh OcH OΔTH ( 0.01kg ) c ( 50 C 100 C) + ( 0.5kg )( 4186 J kg C)( 50 C 20 C) ( 0.5) c J 0 metal metal c metal Q water + Q metal J 0 kg C O iron

13 13 Specific Heat of Ideal Gases The average kinetic energy of a molecule in an ideal gas is K tr 3 2 kt. And the total kinetic energy of the gas is 3 K tr NkT nrt.

14 14 Define the molar specific heat at constant volume; this is the heat capacity per mole. C V Q nδt Heat is allowed to flow into a gas, but the gas is not allowed to expand. If the gas is ideal and monatomic, the heat goes into increasing the average kinetic energy of the particles. 3 Then, the amount of added heat is ΔKtr Q nrδt. 2

15 15 In this situation we can calculate the heat capacity with constant volume for ideal gas: C 3 nrδt Q 2 3 nδt nδt 2 R V 12.5 J/K/mol If the gas is diatomic: C 5 R 2 V 20.8 J/K/mol

16 16 Internal energy will be distributed equally among all possible degrees of freedom (equipartition of energy). Each degree of freedom contributes ½kT of energy per molecule and ½R to the molar specific heat at constant volume. Rotational motions of a 2-atom molecule:

17 Example: A container of nitrogen gas (N 2 ) at 23 C contains 425 L at a pressure of 3.5 atm. If 26.6 kj of heat are added to the container, what will be the new temperature of the gas? 17 For a diatomic gas, Q nc ΔT V. PV RT The number of moles n is given by the ideal gas law i i n. The change in temperature is i ΔT Q C V RTi PiV R 21 K 3 i J R ( 296 K) ( 3.5 atm)( N/m /atm)( 425 L)( 10 m /L) The final temperature of the gas is T f T i + ΔT 317 K 44 C.

18 Phase Transitions ICE WATER STEAM Add heat Add heat These are three states of matter (plasma is another one)

19 Phase Changes A phase change occurs when the physical characteristics of the substance change from one form to another Common phases changes are Solid to liquid melting Liquid to gas boiling Phases changes involve a change in the internal energy, but no change in temperature

20 Phase Transitions 20

21 Latent Heat During a phase change, the amount of heat is given as Q m L L is the latent heat of the substance. Latent heat is the amount of heat per unit mass required to change the phase of a substance. The energy is used to form/break chemical bonds. Latent means hidden or concealed Choose a positive sign if you are adding energy to the system and a negative sign if energy is being removed from the system Latent heat of fusion (L f ) is used for melting or freezing Latent heat of vaporization (L v ) is used for boiling or condensing

22 Problem-solving hints: Use consistent units Transfers in energy are given as QmcΔT for processes with no phase changes Use Q m L f or Q m L v if there is a phase change In Q cold - Q hot be careful of sign, ΔT T f - T i

23 Example: A 75 g cube of ice at C is placed in kg of water at 50.0 C in an insulating container so that no heat is lost to the environment. Will the ice melt completely? What will be the final temperature of this system? 23 The heat required to completely melt the ice is Q ice m c ΔT + m L ice ( kg)( 2.1kJ/kg C)( 10 C) + ( kg)( kj/kg) 27 kj ice ice ice f The heat required to cool the water to the freezing point is Q w m c ΔT w w ( 0.5 kg)( kj/kg C)( 50 C) 105 kj w Since Q ice < Q water the ice will completely melt.

24 24 What will be the final temperature of this system? To find the final temperature of the system, note that no heat is lost to the environment; the heat lost by the water is gained by the ice. 0 Q ice +Q w 0 m ice c ice ΔT + m ice L f + m ice c ( w T f T ) ice,i + m w c ( w T f T ) w,i 0 m ice c ice ΔT + m ice L f + ( m ice + m w )c w T f m w c w T i 0 27 kj + ( m ice + m w )c w T f 105 kj T f 32.4 C

25 25 Example: Compute the heat of fusion of a substance from these data: kj will change kg of the solid at 21 C to liquid at 327 C, the melting point. The specific heat of the solid is kj/kg K. Q L f mcδt + ml Q mcδt m f 22.8 kj/kg

26 26 Phase Diagram On a phase diagram, the triple point is the set of P and T where all three phases can coexist in equilibrium. Sublimation is the process by which a solid transitions into a gas (and gas solid).

27 27 Phase Diagram The critical point marks the end of the vapor pressure curve. A path around this point (i.e. the path does not cross the curve) does not result in a phase transition. Past the critical point it is not possible to distinguish between the liquid and gas phases.