How does calculus relate to chemistry?

Question

Charles Arocha · Accepted Answer

Although this is a fairly general question, I can offer you two instances. Calculus is typically related to chemistry when working with kinetics and thermodynamics. Entropy, a measure of disorder in a system, is one basic example of thermodynamics. According to Physical Chemistry, entropy is defined as follows: #dS = (deltaq_(rev))/(T)# (Eq. A)
where S is entropy, q is heat flow, and T is temperature. At a constant pressure and temperature (such as during a phase transition), it can be written like this:
#dS = (deltaq_(rev,p))/(T) = (dH)/(T)# (Eq. B)
where H is enthalpy. Remember that we're at constant pressure. Now, if you take the partial derivative of entropy with respect to temperature at a constant pressure for Eq. B, you've got the General Chemistry definition of heat capacity---capacity for heat flow at a constant pressure! This is written as #C_P(T)#. You get:
#((deltaS)/(deltaT))_P = ((deltaH)/(deltaT))_P1/T = (C_P(T))/T# (Eq. C-1) If you rewrite this further and take the integral, you get:
#dS = (C_P(T))/TdT#  (Eq. C-2) 
or
#DeltaS = S(T_2) - S(T_1) = int_(T_1)^(T_2)(C_P(T))/TdT# (Eq. D) So if you know the specific heat capacity, then for example, you can figure out its entropy at #500K# if you know its entropy at room temperature (#298K#) and you assume the heat capacity varies negligibly within the temperature range.  The standard state entropy of reaction (#DeltaS_R^o#) can be used for this, for instance. Another example that is a little more complex is when you derive half lives from rate laws in kinetics (this is something you might encounter in Physical Chemistry or AP Chemistry). For example, you could begin with the following "classic" decomposition reaction: #N_2O_4 -> 2NO_2# Since there is only one reactant, you know it is first order. You can write the rate law for this as:
#r(t) = 1/2(d[NO_2])/(dt) = -cancel(1/1)(d[N_2O_4])/(dt) = k[N_2O_4]# If you rearrange this and separate variables, you get:
#-1/([N_2O_4])d[N_2O_4] = kdt# then if you integrate this:
#-int_([N_2O_4]_0)^([N_2O_4])1/([N_2O_4])d[N_2O_4] = int_(t_0)^(t)kdt# you get this:
#-ln(([N_2O_4])/([N_2O_4]_0)) = kt - k(0) = kt#
where #t_0# is 0 seconds. So the half life is then:
#-ln(([N_2O_4]_(1/2))/([N_2O_4]_0)) = -ln ((1/2)/1) = ln2 = kt_(1/2)# #t_(1/2) = ln2/k# Thus, you would infer from this that the half life of a material that is disintegrating on its own (like first-order radioactive decay) is independent of its starting concentration.

William Abdalla · Answer

undefined Calculus is used in chemistry to understand and analyze various chemical processes and phenomena. It helps in determining rates of reactions, modeling chemical systems, and predicting behavior. Calculus is particularly useful in studying kinetics, thermodynamics, and equilibrium in chemistry. It allows for the calculation of reaction rates, rate laws, and rate constants. Additionally, calculus is employed in understanding the behavior of chemical compounds under changing conditions, such as temperature, pressure, and concentration. It helps in solving differential equations that describe chemical reactions and in integrating functions to determine quantities like heat transfer and work done. Overall, calculus provides the mathematical tools necessary for a deeper understanding of chemical principles and their applications.