CHM/PHY 340 Lab
THE THERMODYNAMICS OF RUBBER BANDS
OBJECTIVES 1. To determine how the entropy of a rubber band changes as the rubber band is stretched. 2. To measure partial derivatives experimentally and test the validity of the cyclic rule of partial differentiation. 3. To determine the number of polymeric chains per unit volume in a rubber band.
INTRODUCTION Rubber belongs to a class of macromolecular substances, called elastic polymers, that have the rather unusual characteristic of being able to recover their original shape following a very large deformation. In order for a material to exhibit rubberlike elasticity, three conditions must be met: (1) the material must consist of polymeric chains, (2) the chains must be joined into a network structure, and (3) the chains must have a high degree of flexibility.1 In natural rubber and in many types of synthetic rubber the network structure is present because the rubber is vulcanized, meaning that the individual linear strands of the polymer are cross-linked via S-S bonds to form a three-dimensional structure:
A study of the properties of rubber provides an excellent illustration of several of the concepts of thermodynamics, including the mathematics of partial derivatives, the Maxwell relations, and the molecular interpretation of entropy. We will determine how the entropy of a rubber band changes as it is stretched. We will also examine the cyclic rule of partial differentiation to see if it is valid in the case of rubber. Finally, we will compare our macroscopic measurements to the predictions of a molecular theory of rubber elasticity to determine the number density of polymer chains in the material.
THEORY Thermodynamics provides us with relationships among different properties of a material. For rubber bands we are most interested in the relationships among the tension or restoring force, f, the length, L, the entropy, S, and the temperature, T. One interesting question raised by the above picture of the molecular structure of rubber has to do with the entropy of the material. Does the entropy of rubber increase or decrease as it is stretched isothermally? To restate this question in mathematical terms, is the partial derivative (∂S/∂L)T a positive or negative quantity? revised 4/20/01 1 In Part A we shall answer this question with two different experiments. The first requires observing the change in temperature as the rubber band is stretched adiabatically. From this
measurement we can obtain the sign of the derivative (∂T/∂L)S. According to the cyclic rule of partial differentiation,
∂T ∂L ∂S = −1,(1) ∂ ∂ ∂ L S S T T L
which can be rearranged to
∂S ∂T ∂S = − . (2) ∂ ∂ ∂ L T L S T L
Since the sign of (∂S/∂T)L.is obviously positive (why?), measurement of (∂T/∂L)S will establish
whether (∂S/∂L)T is positive or negative.
A second way of determining the sign of (∂S/∂L)T involves the observation of the temperature
dependence of the length of the rubber band under the condition of constant tension, (∂L/∂T)f. To show the latter quantity is related to the former, we apply the first law of thermodynamics,
dU = dq + dw, (3)
to the deformation or heating of a rubber band. For a reversible process the amount of heat
absorbed is TdS. In addition to expansion work, (−pdV), work equal to fextdL is done on the rubber band when it is stretched a distance dL by an applied force fext. Since fext = f for a reversible process, and since under most circumstances pdV is much smaller than fdL, we can approximate the work as fdL, which permits us to write
dU = TdS + fdL. (4)
Although the above equation was derived for a reversible process it is applicable to all processes, since all the variables in the equation are state functions. Similarly, the infinitesimal change in the Helmholtz free energy, A = U − TS, may be written
dA = −SdT + fdL (5)
An important result is contained in the Maxwell relation derived from eq. (5):
revised 4/20/01 2 ∂S ∂f = − .(6) ∂ ∂ L T T L
Eq. (6) expresses the dependence of entropy on length under isothermal conditions in terms of a more easily measured quantity, the temperature dependence of the restoring force at constant length. The latter is related to the temperature dependence of the length of the rubber band under
the condition of constant tension, (∂L/∂T)f, and the isothermal dependence of tension upon
length, (∂f/∂L)T, by the cyclic rule:
∂f ∂T ∂L ⋅ ⋅ = −1.(7) ∂ ∂ ∂ T L L f f T
Substition of (6) into (7), followed by rearrangement, yields
∂S ∂L ∂f = ⋅ . (8) ∂ ∂ ∂ L T T f L T
Since (∂f/∂L)T is positive (why?), a determination of the sign of (∂L/∂T)f, will establish the sign
of (∂S/∂L)T.
Given that our determination of the sign of partial derivative (∂S/∂L)T is so greatly dependent on the cyclic rule, it is important to demonstrate that the cyclic rule is valid. In Part B of the experiment you will investigate the validity of eq. (7) by carrying out measurements to determine
each of the partial derivatives independently. For example, the derivative (∂L/∂T)f is equal to the slope of a plot of L vs. T for a series of measurements in which the tension is fixed at a certain value. In setting up your experiment it will be helpful to note which variable is the dependent one, whose value you measure but do not choose, and which variables are independent, the values of which you must choose. Note also that in each case one of the two independent variables is being held constant and one is being varied.
Finally, we shall use our data to learn about the density of polymer chains in the rubber band. According to the simplest molecular theories of rubber elasticity,1 the restoring force follows the relationship
1 f = nAkα − T , (9) α 2 where n = number of polymer chains per unit volume A = cross-sectional area of the unstretched rubber band
revised 4/20/01 3 k = Boltzmann constant
α = L/L0, the ratio of the length to the unstretched length.
Therefore, the value of n can be determined from the slope of f vs. T, i.e. (∂f/∂T)L, at a particular value of L.
APPARATUS The apparatus for Part B is shown in Figures 1 and 2. The sample is suspended in a vacuum- jacketed condenser tube. The sample is heated by a stream of N2 that is passed through an ice bath to cool it below room temperature and then over a resistance heater contained in another insulated condenser. A temperature controller senses the temperature via a thermocouple and controls the current to the heater.
The length of the sample is set by adjusting the position of a small lab jack that is mounted on the platform of a digital balance. A cathetometer is used to measure the elongation of the rubber band. The difference between readout of the balance with and without the rubber band attached to the lab jack equals the tension of the rubber band.
PROCEDURE Part A. Qualitative measurements 1. Determination of the sign of (∂T/∂L)S. Place your thumbs through a heavy rubber band, one on each end. Without stretching the band, hold it to your forehead or upper lip. Note whether the rubber band feels cool or warm. Next move the rubber band slightly away from your face so that it is not touching your skin. Quickly stretch the rubber band as far as you can and, holding it in the stretched position, touch it again to your forehead or lip. Note whether the temperature of the rubber band has increased, decreased, or remained unchanged compared to the unstretched band. Move the stretched rubber band away from your face. Quickly let it relax to its original size, and again hold it to your skin. Note whether the temperature has increased, decreased, or remained unchanged compared to the stretched band.
The stretching and unstretching processes are approximately adiabatic because they occur quickly. Therefore, they are isoentropic processes. Use your observations to determine the sign of (∂T/∂L)S.
2. Determination of the sign of (∂L/∂T)f. Hang one end of the rubber band from a hook, and suspend a weight from the other end. The weight should be heavy enough to stretch the rubber band, but not so heavy that it is likely to break it. Heat the rubber band with a hair dryer. Note whether the rubber band becomes longer or shorter when it is heated. Turn off the heater and allow the rubber band to cool. Note whether its length changes as it cools.
The tension in the rubber band must equal the suspended weight, so f is constant throughout the heating and cooling. Use your observations to determine the sign of (∂L/∂T)f.
revised 4/20/01 4 Part B. Quantitative Measurements 3. A sample of the rubber band will be provided. Using a micrometer, measure the two smallest dimensions the sample, being careful not to compress it. Determine the cross-sectional area. 4. You must begin by taring the balance and finding the position of the cathetometer that corresponds to zero elongation: Set the lab jack to its highest position and switch on the balance. Adjust the position of the rubber band so that it is just beginning to stretch. (It may be easier to tell if the rubber band is stretching if you first loosen the set screw coupling the sample rod to the lab jack so that there is some slack in the system.) Tare the balance, and use the cathetometer to read the position of the marker on the rod. Gently tighten the set screw if you loosened it earlier, being careful not to jar or misalign the sample rod. The balance should still read very close to zero. If it doesn’t, check the alignment of the sample rod, and retare the balance if necessary. 5. Open the regulator for the nitrogen gas, and allow the gas to flow through the apparatus at a rate of about 5 L/min. (Use a flowmeter to measure the rate.) Obtain a value of L0 by measuring the length of the unstressed rubber band using a ruler held behind the sample. 6. Measurement of (∂f/∂T)L.at α = 2. a. Gently lower the lab jack so that the sample length is doubled (L/L0 ≈ 2). You will notice that the balance indicates a negative weight. b. Set the temperature controller to the desired temperature. (A good starting point is about 25°C.) Do not let the heater current exceed 1 A. Wait for a few minutes after the set point is reached, then adjust the lab jack so that the L = 2⋅L0. Record the temperature and the balance reading after they have stabilized. c. Repeat step b for enough other temperatures so that you have sufficient data to find (∂f/∂T)L. Don’t go over 80°C. If possible, also use a temperature below room temperature. 7. Measurement of (∂L/∂T)f. Devise and carry out a series of measurements that will allow you to determine (∂L/∂T)f in the vicinity of 25°C, α = 2. Choose the constant value of f to be equal its value for 25°C and α = 2. 8. Measurement of (∂f/∂L)T. Devise and carry out a series of measurements that will allow you to determine (∂f/∂L)T. at 25°C in the vicinity of α = 2.
7. Shut off the temperature controller, balance, and gas flow when you are finished.
DATA ANALYSIS Part A. Qualitative determination of the sign of (∂S/∂L)T. 1. Use your result for the sign of (∂T/∂L)S to determine the sign of (∂S/∂L)T. Use your result for the sign of (∂L/∂T)f to determine the sign of (∂S/∂L)T. If the signs of (∂S/∂L)T don’t agree, you should check your analysis or repeat your qualitative measurements!
Part B. Quantitative test of the cyclic rule. 2. (∂f/∂T)L.at α = 2, T = 25°C. Use the data in step B6 to plot f vs. L. This plot should be linear (see eq. 9), so find the slope of the best line through the points.
3. (∂L/∂T)f. α = 2, T = 25°C. Use the data in step B7 to plot L vs. T. If the data are linear, find the slope of the line. If they are nonlinear, fit the data with the quadratic function L = a + bT + cT2, and find the revised 4/20/01 5 coefficients a, b, and c. Take the derivative of the quadratic to determine an equation for the slope of L vs. T, and evaluate the slope at 25°C.
4. (∂f/∂L)T. at α = 2, T = 25°C. Use the data in step B8 to plot f vs. L. You will probably find that this graph is nonlinear. There are two choice for dealing with this. You may fit the data to a quadratic expression and take the derivative to find the slope at L = 2L0. Alternatively, eq. (9) shows that a plot of − f vs. R, where R = α − α 2, should be a straight line. According to the chain rule of calculus,
∂f ∂f ∂R ∂α = ⋅ ⋅ . (10) ∂ ∂ ∂α ∂ L T R T T L T
The first quantity on the right hand side is the slope of f vs. R. The second and third quantities can be determined by analytical differentiation of the expressions for R and α, and then evaluated for L = 2L0.
5. Use your results to evaluate the left side of the cyclic rule [equation (7)] as well as the uncertainty in the product of the partial derivatives. A more convenient form of equation (7) is
∂ ∂ L ⋅ f ∂T ∂L f T = −1, (7a) ∂ f ∂ T L
6. Use eq. (9) and your value of (∂f/∂T)L to determine n, the number of polymer chains per cubic meter. Make sure that you express (∂f/∂T)L in N/K rather than g/K, otherwise the units won’t cancel properly!
DISCUSSION Examine the molecular picture of cross-linked rubber shown above. Give a molecular argument to account for the sign of (∂S/∂L)T.
Can you conclude that the cyclic rule is valid to within experimental uncertainty?
REFERENCES 1 Mark, J. E. J. Chem. Educ., 1981, 58, 898.
revised 4/20/01 6 cathetometer
thermocouple
FIGURE 1. APPARATUS USED TO VARY LENGTH, TENSION, AND TEMPERATURE OF A RUBBER BAND
revised 4/20/01 7 FIGURE 2. ELECTRICAL CONNECTIONS TO REAR PANEL OF OMEGA TEMPERATURE CONTROLLER
Type K thermocouple
yellow red
+ - TC ANALOG OUT
115 V AC (line)
NC NO COM
DO NOT EXCEED A 1 A!
variable transformer “Variac”
heating wire
revised 4/20/01 8