GATE CY 2018 Question Paper (04-Feb-2018) (Shift 2)

At temperature T, the canonical partition function of a harmonic oscillator with fundamental frequency ($\nu$) is given by
$q_{vib}(T)=\frac{e^{-h\nu /2k_{B}T}}{1-e^{-h\nu /k_{B}T}}$
For $\frac{h\nu }{k_{B}T}=3$, the probability of finding the harmonic oscillator in its ground vibrational state is ____________ (Up to two decimal places)

The enthalpy of vaporization of a liquid at its boiling point ($T_b=200\text{ K}$) is $15.3{\text{ kJ mol}}^{-1}$. If the molar volumes of the liquid and the vapour at 200 K are 110 and $12000{\text{ cm}}^3{\text{ mol}}^{-1}$ respectively, then the slope $\frac{dP}{dT}$ of the liquid-vapour boundary is ____________ kPa $K^{-1}$. (Upto two decimal places. Note: $1\text{ Pa}=1{\text{ J m}}^{-3}$)

In a molecule XY, let ${\psi }_X$ and ${\psi }_Y$ denote normalized atomic orbitals of atoms X and Y,respectively. A normalized molecular orbital of XY is given by ${\psi }_s=0.56({\psi }_X+{\psi }_Y)$. The value of the overlap integral of ${\psi }_X$ and ${\psi }_Y$ is ____________ (Up to two decimal places)

The absorption maxima of two dyes X and Y are 520 and 460 nm, respectively. The absorbance data of these dyes measured in a 1 cm path length cell are given in the table below.The concentration of Y in the mixture is ____________ mM. (Up to two decimal places)

The $\pi$ electrons in benzene can be modelled as particles in a ring that follow Pauli's exclusion principle. Given that the radius of benzene is 1.4 Å, the longest wavelength of light that is absorbed during an electronic transition in benzene is ____________ nm. (Up to one decimal place. Use $m_e=9.1\times 10^{-31}\text{ kg}$, $h=6.6\times 10^{-34}\text{ Js}$, $c=3.0\times 10^8{\text{ m s}}^{-1}$)