#$&* course PHY 202 2/15 about 2 pm query problem 15 introductory problem sets temperature and volume information find final temperature.
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Given Solution: ** PV = n R T so n R / P = V / T Since T and V remain constant, V / T remains constant. • Therefore n R / P remain constant. • Since R is constant it follows that n / P remains constant. ** STUDENT QUESTION: I don’t understand why P is in the denominator when nR was moved to the left side of the equation INSTRUCTOR RESPONSE: The given equation was obtained by dividing both sides by P and by T, then reversing the sides. We could equally well have divided both sides by v and by n R to obtain P / (n R) = T / V, and would have concluded that P / n is constant. To say that P / n is constant is equivalent to saying the n / P is constant. Your Self-Critique:OK Your Self-Critique Rating:OK ********************************************* Question: why, in terms of the ideal gas law, is T / V constant when only temperature and volume change? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: PV=nRT T/V = P/(nR), If only T and V cange, the number of moles and Pressure are remaining the same, as given. Since the right side of the equation doesn’t change, since R is a constant, the ratio T/V must remain the same, even if T and V are changing. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** STUDENT ANSWER AND INSTRUCTOR RESPONSE: They are inversely proportional. They must change together to maintain that proportion. INSTRUCTOR RESPONSE: You haven't justified your answer in terms of the ideal gas law: PV = n R T so V / T = n R / P. If only T and V change, n and P don't change so n R / P is constant. Therefore V / T is constant, and so therefore is T / V. You need to be able to justify any of the less general relationships (Boyle's Law, Charles' Law, etc.) in terms of the general gas law, using a similar strategy. ** Your Self-Critique:OK Your Self-Critique Rating:OK ********************************************* Question: prin phy Ch 13.26. Kelvin temperatures corresponding to 86 C, 78 F, -100 C, 5500 C and -459 F. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: 86 C + 273.15 = 359.15 K or 359 K (5/9)*(78 F - 32) = 26 C, 26 C + 273.15 = 299 K -100 C + 273.15 = 173.15 K or 173 K 5500 C + 273.15 = 5773.15 K or 5773 K (5/)*(-459 F - 32) = -273 C, -273 C + 273.15 = 0.15 K or 0 K confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The Kelvin temperature is 273 K higher than the Celsius temperature (actually 273.15 below, but the degree of precision in the given temperatures is not sufficient to merit consideration of tenths or hundredths of a degree). • 86 C, -100 C and 5500 C are therefore equivalent to ( 86 + 273 ) K = 359 K, -100 + 273 K = 173 K, (5500 + 273) K = 5773 K. The freezing point of water is 0 C or 32 F, and a Fahrenheit degree is 5/9 the size of a Celsius degree. Therefore • 78 F is (78 F - 32 F) = 46 F above the freezing point of water. • 46 Fahrenheit degrees is the same as (5/9 C / F ) * 46 F = 26 C above freezing. • Since freezing is at 0 C, this means that the temperature is 26 C. • The Kelvin temperature is therefore (26 + 273) K = 299 K. Similar reasoning can be used to convert -459 F to Celsius • -459 F is (459 + 32) F = 491 F below freezing, or (5/9 C / F) * (-491 F) = 273 C below freezing. • This is -273 C or (-273 + 273) K = 0 K. • This is absolute zero, to the nearest degree. Your Self-Critique:OK Your Self-Critique Rating:OK ********************************************* Question: prin phy and gen phy Ch 13.30 air at 20 C is compressed to 1/9 of its original volume. Estimate the temperature of the compressed air assuming the pressure reaches 40 atm. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: PV=nRT, nR are constant so PV/T = nR, so we have P_1*V_1/T_1 = P_2*V_2/T_2 and we need to find T_2 so T_2 = (P_2*V_2*T_1)/(P_1*V_1) = (40 atm * (1/9) * (20 C +273.15))/(1 atm * 1) = 40 atm * (1/9) * 293 K = 1302 K confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: First we reason this out intuitively: If the air was compressed to 1/9 its original volume and the temperature didn’t change, it would end up with 9 times its original pressure. However the pressure changes from 1 atm to 40 atm, which is a 40-fold increase. The only way the pressure could end up at 40 times the original pressure, as opposed to 9 times the original, would be to heat up. Its absolute temperature would therefore have to rise by a factor of 40 / 9. Its original temperature was 20 C = 293 K, so the final temperature would be 293 K * 40/9, or over 1300 K. Now we reason in terms of the ideal gas law. P V = n R T. In this situation the number of moles n of the gas remains constant. Thus P V / T = n R, which is constant, and thus P1 V1 / T1 = P2 V2 /T2. The final temperature T2 is therefore • T2 = (P2 / P1) * (V2 / V1) * T1. From the given information P2 / P1 = 40 and V2 / V1 = 1/9 so • T2 = 40 * 1/9 * T1. The original temperature is 20 C = 293 K so that T1 = 293 K, and we get • T2 = 40 * 1/9 * 293 K, the same result as before. Your Self-Critique:OK Your Self-Critique Rating:OK ********************************************* Question: query gen phy ch 13.38 fraction of air released after tire temp increases from 15 to 38 C at 220 kPa gauge YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: T1 = 15 C, 15 C + 273.15 = 288.15 K T2 = 38 C, 38 C + 273.15 = 311.15 K P_g = 220 kPa so P = 220 kPa + 101 kPa = 321 kPa nRT is constant so we have P1/T1 = P2/T2 P2 = (P1 * T2)/P2 = (321 kPa * 311.15 K)/288.15 K = 347 kPa Heating raises the pressure to 347 kPa Releasing the air to return P back to 321 kPa, the number of moles will change along with the pressure while everything else will assume to be constant. PV=nRT, P/n=RT/V P2/n2 = P3/n3 N2/n3 = P2/P3 This ratio for the decrease in P will show the ratio of the decrease in moles since all others are constant. P2/P3 = 347 kPa / 321 kPa = 0.93 Since there is a 7% decrease in pressure, so is the decrease in the number of moles or amount of air released. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: (Note that the given 220 kPa initial gauge pressure implies an absolute pressure of 311 k Pa; assuming atmospheric pressure of about 101 k Pa, we add this to the gauge pressure to get absolute pressure). Remember that the gas laws are stated in terms of absolute temperature and pressure. The gas goes through three states. The temperature and pressure change between the first and second states, leaving the volume and the number n of moles constant. Between the second and third states pressure returns to its original value while volume remains constant and the number n of moles decreases. From the first state to the second: T1 = 288 K, T2 = 311 K so T2 / T1 = 311 / 288 = 1.08, approx. This is approx. an 8% increase in temperature. The pressure must therefore rise to P2 = 3ll / 288 * 321 kPa = 346 kPa, approx (note that we have to use actual rather than gauge pressure so init pressure is 220 kPa + 101 kPa = 321 kPa, approx. ) From the second state to the third, pressure is then released by releasing some gas, changing the number n of moles of gas in order to get pressure back to 331 kPa. Thus n3 / n2 = P3 / P2 = 321 kPa / 346 kPa or approximately .93, which would be about a 7% decrease. So we have to release about 7% of the air. Note that these calculations have been done mentally, and they might not be particularly accurate. Work out the process to botain the accurate numerical results. Note also that temperature changes from the second to third state were not mentioned in the problem. We would in fact expect a temperature change to accompany the release of the air, but this applies only to the air that escapes. The air left in the tire would probably change temperature for one reason or another, but it wouldn't do so as a direct result of releasing the air. STUDENT QUESTION It seems that the air goes from 288 to 311 K, so the ratio should be n2 / n1 = 288 / 311 and the proportional loss should be about (1 - 288 / 311) INSTRUCTOR RESPONSE The Kelvin temperature goes from 288 K to 311 K. If the air is released at constant pressure, then volume and pressure remain constant while temperature and number of moles vary according to n T = P V / R so that n1 T1 = n2 T2, and n2 = n1 * (T1 / T2) = n1 * (288 / 311) and the change in amount of gas is n1 - n2 = n1 - 288/311 n1 = n1( 1 - 288 / 311), or about 7.4% of n1. If the temperature is first raised to 311 K, then the gas is released, it is the pressure and amount of gas that change. In that case the change in the amount of the gas is n1 - n3 = n1 - (321 kPa / 346 kPa) * n1 = n1 ( 1 - 321 / 346), or about 7.2% of n1. The fractions 288/311 and 321 / 346 don't differ by much, not do the percents, but they do differ. Your Self-Critique:OK Your Self-Critique Rating:OK ********************************************* Question: `q001. The temperature of a certain object increases from 50 Celsius to 150 Celsius. What is its change in temperature in Celsius? Convert 50 Celsius and 150 Celsius to Kelvin. What is the change in temperature in Kelvin? What is the change in temperature in Fahrenheit? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: ‘dT = 150 C - 50 C = 100 C T1 = 50 C, 50 C + 273.15 = 323.15 K T2 = 150 C, 150 C + 273.15 = 423.15 K ‘dT_K =423.15 K - 323.15 K = 100 K T1 = 50 C, (9/5)*50 + 32 = 122 F T2 = 150 C, (9/5)*150 + 32 = 302 F ‘dT_F = 302 F - 122 F = 180 F confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-Critique Rating:OK ********************************************* Question: `q002. A sample of gas originally at 0 Celsius and pressure 1 atmosphere is compressed from volume 450 milliliters to volume 50 milliliters, in which state its pressure is 16 atmospheres. What is its temperature in the new state? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: PV=nRT, n and R constant so PV/T = nR P1V1/T1 = P2V2/T2 T2 = (P2V2T1)/(P1V1) = (16 atm * 50 mL * (0 + 273.15)K)/(1 atm * 450 mL) = 218520 (atm*mL*K) / 450 atm*mL) = 485.6 K confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-Critique Rating:OK ********************************************* Question: `q003. What product or ratio involving P, V, n and T would remain constant if V and T were held constant? Why does this make sense? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: V/T = n*R / P, or V = constant * T, since R is always constant, n/P ration would remain constant if V and T are held constant. The ways to increase P are decrease volume, increase T or increase n. If V and T are constant, n/P ration must remain the same, even if n or P changes. This would mean that if n changes, P must change to hold the same ratio, so if n increases, P increase. This would make sense because if you increase the gas in the space, the pressure must rise to keep the ratio the same, n/P. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-Critique Rating:OK ********************************************* Question: Openstax: Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of 10.0ºC and rises to an altitude where its volume is twenty times the original volume and its temperature is - 50.0ºC ? (b) What is the gauge pressure? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: T1 = 10 C, 10 C + 273.15 = 283.15 K T2 = -50 C, -50 C + 273.15 = 223.15 K V1 = x V2 = 20x P1 = y atm P2 = ? PV = nRT, R and n constant PV/T = nR P1V1/T1 = P2V2/T2 P2 = (P1V1T2)/(T1V2) = y atm * x * 223.15 K / (283.15 K * 20x) = (223.15 K * y atm) / (5663 K) = 0.0394 * y or 0.039 * P1 I needed help from the given solution for this part. Pressure of gauge, assuming 1 atm at the surface, would be 0.039 atm - 1 atm = -0.961 atm confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Simple reasoning: the volume increases by a factor of 20, which by itself would imply that the pressure decreases by a factor of 20. But absolute temperature also decreases by a factor of 223 / 283, which would also imply a pressure decrease. Bottom line: The pressure changes by factor (1/20) * (223/283) = .039, ending up at .039 of its original value. More reliable analysis: If V_0 and V_f are the initial and final volumes, then V_f / V_0 = 20 The initial and final temperatures are 10 C = 283 K and -50 C = 223 K, so the ratio of temperatures is T_f / T_0 = 223 / 283. Since no helium escapes, n is constant to the PV = n R T implies that P V / T is constant. Thus P_0 V_0 / T_0 = P_f V_f / T_f. Solving this for the pressure P_f we obtain P_f = P_0 * V_0 / V_f * T_f / T_0 = P_0 * (1/20) * (223/283) = P_0 * .039. That is, the pressure at the new altitude is about .039 that at the surface. Assuming the surface pressure to have been 1 atmosphere, the gauge pressure at altitude would be (.039 atm - 1 atm) = -.961 atm. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):I was unsure of the gauge pressure at first but I do understand now. ------------------------------------------------ Self-critique Rating:3 #$&* ********************************************* Question: (a) What is the average kinetic energy in joules of hydrogen atoms on the 5500ºC surface of the Sun? (b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is 6.00×10^5 K ? The average kinetic energy of particles at temperature T is KE_ave = 3/2 k T, where k = R / N_A is the Boltzmann constant (R is the gas constant and N_A is avagodro's number). The average kinetic energy of a particle does not depend on what kind of particle it is. However the less massive the particle, the greater will be its speed at the average kinetic energy. At 5500 C the absolute temperature is 5773 K so the average KE of a particle is KE_ave = 3/2 k T = 3/2 * 1.38 * 10^-23 Joules / (particle Kelvin) * 5773 K = 1.19 * 10^-19 Joules. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: T=5500 C = 5773.15 K KE_ave = (3/2)*k*T = (3/2)*(1.38 * 10^-23 J/particle K)*(5773.15 K) = 1.20 * 10^-19 J T = 6.00*10^5 K KE_ave = (3/2)*k*T = (3/2)*(1.38 * 10^-23 J/particle K)*(6.00 * 10^5 K) = 1.24 * 10^-17 J confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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