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## Homework 11

General Physics 2, Spring '10

Homework 11

This will be the final assignment of the semester. It consists of a couple of questions from Chapter 10, and then an analysis/write-up of the experiment (to measure Boltzmann's constant and so indirectly Avogadro's number) we'll do in class either Friday or Monday. You don't need to turn this in until Wednesday the 5th (which is the last day of classes), but perhaps you want to work on the non-lab-based questions for Friday so as to avoid a heavy workload next Tuesday night.

1. QTD (not Project) 2: "Consider a bunch of gas molecules..."

2. QTD (not Project) 5: "Consider the gas, initially confined..."

3. As you might know, you have to add heat to ice to make it melt. But during this melting process, the temperature stays constant -- the added energy just goes into breaking up the bonds that hold the water molecules together in the solid phase. Quantitatively, the so-called "heat of fusion" of water is about 80 calories/gram. This just means that 80 calories of heat must be added to one gram of ice (at zero degrees C) to convert it into one gram of liquid water (at zero degrees C). That's background. Here's the question. Imagine a thermally insulated bucket of pure liquid water (no ice) at zero degrees C. In principle, it should be possible for the energy to spontaneously re-distribute itself among the water molecules such that a one-gram ice cube forms (at the expense of the rest of the water getting a teensy bit warmer). What is the probability of such an event happening spontaneously? (Hint: use the old Clausius definition of entropy to determine the decrease in entropy associated with the formation of a one gram ice cube; then use Boltzmann's statistical interpretation of entropy to convert this into a probability for the event. You'll need to use Boltzmann's constant k_B for this one. You can either use the value you get from the experiment, or you can use the nominal value: k_B = 1.4 * 10^(-23) Joules/Kelvin.)

4. By carefully counting dots in the microscope images captured in class, determine a value (with uncertainty) for Boltzmann's constant "k_B". Then, using the known value of the ideal gas constant "R" (8.3 Joules/mole*Kelvin), convert this into a value (with uncertainty) for Avogadro's number "N_A". Note that this amounts to doing a curve fit using Equation (10.100) from the book. It will probably be helpful, though, to think about making a new equation by taking the logarithm of both sides of 10.100, which results in: log( n( h ) ) = constant - alpha * h. That is: a graph of the logarithm of n vs. h should be a straight line with slope -alpha, so one can make such a graph and just find the range of slopes which fits the data well. Then one can use Equation 10.101 to relate the (now known) value of alpha to the desired thing, k_B.

Notes:

Homework 11

This will be the final assignment of the semester. It consists of a couple of questions from Chapter 10, and then an analysis/write-up of the experiment (to measure Boltzmann's constant and so indirectly Avogadro's number) we'll do in class either Friday or Monday. You don't need to turn this in until Wednesday the 5th (which is the last day of classes), but perhaps you want to work on the non-lab-based questions for Friday so as to avoid a heavy workload next Tuesday night.

1. QTD (not Project) 2: "Consider a bunch of gas molecules..."

2. QTD (not Project) 5: "Consider the gas, initially confined..."

3. As you might know, you have to add heat to ice to make it melt. But during this melting process, the temperature stays constant -- the added energy just goes into breaking up the bonds that hold the water molecules together in the solid phase. Quantitatively, the so-called "heat of fusion" of water is about 80 calories/gram. This just means that 80 calories of heat must be added to one gram of ice (at zero degrees C) to convert it into one gram of liquid water (at zero degrees C). That's background. Here's the question. Imagine a thermally insulated bucket of pure liquid water (no ice) at zero degrees C. In principle, it should be possible for the energy to spontaneously re-distribute itself among the water molecules such that a one-gram ice cube forms (at the expense of the rest of the water getting a teensy bit warmer). What is the probability of such an event happening spontaneously? (Hint: use the old Clausius definition of entropy to determine the decrease in entropy associated with the formation of a one gram ice cube; then use Boltzmann's statistical interpretation of entropy to convert this into a probability for the event. You'll need to use Boltzmann's constant k_B for this one. You can either use the value you get from the experiment, or you can use the nominal value: k_B = 1.4 * 10^(-23) Joules/Kelvin.)

4. By carefully counting dots in the microscope images captured in class, determine a value (with uncertainty) for Boltzmann's constant "k_B". Then, using the known value of the ideal gas constant "R" (8.3 Joules/mole*Kelvin), convert this into a value (with uncertainty) for Avogadro's number "N_A". Note that this amounts to doing a curve fit using Equation (10.100) from the book. It will probably be helpful, though, to think about making a new equation by taking the logarithm of both sides of 10.100, which results in: log( n( h ) ) = constant - alpha * h. That is: a graph of the logarithm of n vs. h should be a straight line with slope -alpha, so one can make such a graph and just find the range of slopes which fits the data well. Then one can use Equation 10.101 to relate the (now known) value of alpha to the desired thing, k_B.

Notes:

- the "effective mass" (for the little plastic balls) that we calculated in class was 2.14 * 10^(-14) grams. Or equivalently: 2.14 * 10^(-17) kg.
- I measured the temperature in the room (where the microscope and cell sat all night) right after capturing the images, and it was T = 296.9 Kelvin.

Last modified: Monday, December 19, 2011, 9:18 AM