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.