# Questions 87 - 96#

## Q87 H$$_2^+$$ molecular orbitals#

This question is about the energy of H$$_2^+$$ molecular orbitals. The definitions in 11.2 are used. If $$\varphi_1$$ and $$\varphi_2$$ are H atom 1S orbitals show that

(a) $$\displaystyle \int\varphi_1 \frac{q^2}{R}\varphi_2 d\tau=\frac{q^2S}{R}$$ where $$R$$ is the internuclear separation.

(b) Calculate the resonance integral $$\displaystyle A=\int\varphi_1 \frac{q^2}{r_2}\varphi_2 d\tau$$ belonging to $$H_{12}$$ and thereby calculate the energy.

(c) Use the results from the text and those just calculated and plot the energies $$E_+$$ and $$E_-$$ with internuclear separation and so reproduce figure 31. Use $$\rho=a_0R, a_0=1$$ and electronic charge $$q=1$$.

## Q88 Arc length#

Calculate the arc length for

(a) a circle of radius $$R$$,

(b) the logarithmic or equiangular spiral $$r = e^{-\theta/a}$$ from $$0 \to 2\pi$$,

(c) the catenary $$y = \cosh(x)$$ from $$x = 0 \to x_0$$, and

(d) the Archimedean spiral $$r = a\theta$$ from $$0 \to 2\pi$$.

Strategy: Use equation 83 or 84.

## Q89 Area#

Find the area, the $$x$$ and $$y$$ centroids and moments of inertia $$I-x, I_y$$, and $$I_z$$ of the ellipse $$x^2 + y^2 = 1$$.

## Q90 Mean value#

Calculate the mean value of $$r^2 = x^2 + y^2$$ over the ellipse defined in the previous question.

## Q91 Line integral#

If $$C$$ is a line joining $$(0, 0)$$ to $$(a, b)$$ calculate $$\displaystyle \int_C e^x\sin(y)dx+e^x\cos(y)dy$$.

Strategy: Use the two function formula and convert $$dy$$ into $$dy/dx$$ where $$y$$ is determined by the limits on the line, in this case a straight line from the origin to $$(a, b)$$.

## Q92 Area#

(a) Find the area under one arch of the cycloid that is described by the parametric equations $$x = a(t - \sin(t)),\, y = a(1 - cos(t))$$. A description and sketch of the cycloid is given in Figure 16.

(b) Find the length of the arch.

## Q93 Arc length#

Calculate the arc length for curves

(a) $$r = 1$$

(b) $$r = e^{-\theta}$$ from $$0 \to 2\pi$$, and (c) the catenary $$y=\cosh(x)$$ from $$x=0\to x_0$$ where $$x_0 \gt 0$$.

## Q94 Surface area#

The surface area of a function $$f(x)$$ is given by $$\displaystyle A=2\pi\int_a^b f(x)\sqrt{1+f'(x)^2}dx$$.

(a) Show that the surface area of a sphere is $$4\pi r^2$$ starting with a circle of radius $$r$$, in which case $$f(x) = r^2 - x^2$$, and effectively rotating this to form the surface. The integration limits are $$\pm r$$.

(b) Work out what fraction of the earth’s surface is north of the seaside town of Dunbar, Scotland that is situated at exactly lat $$56.00^\mathrm{o}$$ N.

Note: $$f'(x)$$ is the first derivative. Latitude is the angle from the equator to the pole.

Strategy: In (a) substitute, simplify, and find a very simple integral. In (b) take the south to north axis of the earth to be the x-axis and work out the $$x$$ integration limits.

## Q95 State function#

(a) In thermodynamics, what is a state variable?

(b) The work $$w$$ required to expand a gas is the line integral $$w = -\int p dV$$. If $$T$$ and $$p$$ are the variables to be used, this equation can be written as

$\displaystyle w =\int p\left(\frac{\partial V}{\partial T}\right)_p dT+p\left(\frac{\partial V}{\partial p}\right)_T dp$

For 1 mole of an ideal gas calculate $$w$$ along each of the two paths used in the example in Section 13.9 and Figure 36 and hence show that $$w$$ is not a state function.

Strategy: Follow the example and make the integral into one in $$dp$$ and then $$dT$$ alone. Substitute for the partial derivatives and use the gas law to substitute variables to make an equation in $$p$$ or $$T$$ as necessary. Only then, work out the remaining derivative, $$dp/dT$$ or $$dT/dp$$ depending on the path taken.

## Q96 Entropy of van der Waals gas#

Calculate the entropy for an van der Walls gas whose equation of state is $$(p+a/V^2)(V-b)=RT$$ where $$a,\,b$$ are constants. Is the entropy different to that of an ideal gas and if so why?

Strategy: Calculate $$(\partial V/\partial T)_p$$ then use equation 89.