Area under a graph

This integral is the area under a graph of 1/V against V (a hyperbola) from Vj to V2. It defines the natural logarithm function, symbolized In (see Appendix C). In particular. 

During an expansion, must be less than F, the pressure of the gas. For a reversible expansion, Fg t is only infinitesimally smaller (so the system is always close to equilibrium) but for an irreversible expansion, F xt is measurably smaller. Therefore, the area under a graph of F xt plotted against V is less than that of a graph of F against V (Fig. 13.7), so... 

Measuring the intercept, gradient and area under a graph... 

The area under a graph of any function/is found by the techniques of integration. For instance, the area under the graph of the function/(x) between x= a and x= bis denoted by... 

If T is normally distributed witli mean p and standard deviation a, then tlie random variable (T - p)/a is normally distributed with mean 0 and standard deviation 1. The term (T - p)/a is called a standard normal variable, and tlie graph of its pdf is called a "standard normal curve. Table 20.5.2 is a tabulation of areas under a standard normal cur e to tlie right of Zo of r normegative values of Zo. Probabilities about a standard normal variable Z can be detennined from tlie table. For example,... 

An integral of a function—in this case, the integral of Cp/T—is the area under the graph of the function. Therefore, to measure the entropy of a substance, we need to measure the heat capacity (typically the constant-pressure heat capacity) at all temperatures from T = 0 to the temperature of interest. Then the entropy of the substance is obtained by plotting CP/T against T and measuring the area under the curve (Fig. 7.11). 

The area under a Maxwell-Boltzmann distribution graph represents the distribution of the kinetic energy of collisions at a constant temperature. At a given temperature, only a certain fraction of the molecules in a sample have enough kinetic energy to react. 

The probability that the variable x takes a value between a and b is given by the area under the graph of the probability distribution between x=a and x=b. This is illustrated in Figure 21.3, where the shaded area gives the probability that the standard normal variate, Z, lies between z and infinity, i.e. the probability P(Z>z). The total area under the graph is equal to 1, and because of the symmetry of the normal distribution it follows that the area of any one half is equal to 0.5. For any normal... 

From the area under the graph up to a = 0.80, we find that the required reaction time is approximately 720 8 = 12 min. 

Over a series of time intervals Af, each sample is collected by allowing a small fraction of the fluid to flow steadily into a little cup these samples are stirred to make them uniform and then analysed. Each sample thus represents the concentration in some way averaged over the interval Af(. When these concentrations are plotted as the stepwise graph shown in Fig. 2.176, the area under the graph is equal to the summation C,Af, but, in order to approximate equations 2.27 and 2.28, the appropriate times at which to take moments (first and second) are not immediately apparent because of the unknown nature of the concentration averaging process. Arbitrarily, we take the mid-increment value t as shown. 

Note that we used Equation 12.13 to replace Pg t with the internal pressure P in this reversible process. This is a compact way of saying that the work, w, for a reversible expansion process is (with a minus sign) the area under the graph of P plotted against V from Vi to V2. 

The reason is self-evident if a function is multiplied everywhere by a constant factor, then the area under its graph must be increased by the same factor. Second, the integral of the sum of two functions is the sum of the separate integrals ... 

Areas under a straight-line graph and a curve graph... 

For the majority of the graphs in chemistry the area under the graph does not represent a useful physical quantity. However, the area under the curve is relevant to the Maxwell-Boltzmann distribution curve (Chapter 6), which is useful in accounting for the rate of reaction at different temperatures. It is a frequency distribution curve which shows the distribution of kinetic energies among reacting gas particles at a particular absolute temperature, T (in kelvins) (Figure 11.37). 

The area under the graph in Figure 11.37 is proportional to the number of gas particles. The graph shows that a certain number of particles with kinetic energies equal to or greater than the activation energy E, are able to undergo reaction. 

All we need to recognize is the standard result from calculus, that the integral of a function between two limits is the area under the graph of the function between the two limits. In this case, the function is C/T, the heat capacity at each temperature divided by that temperature, and it follows that... 

C/r against T, and evaluate the area under the graph between the temperatures and Tf. In practice, mathematical software is used to fit a curve to the variation of C with T and the integration carried out automatically. 

I I 4. In general, the entropy change accompanying the heating of a system is equal to the area under the graph of C/T against Tbetween the two temperatures of interest. 

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