The heat of atomisation of methane and ethane are \( x \) kJ mol\(^{-1}\) and \( y \) kJ mol\(^{-1}\) respectively. The longest wavelength (\( \lambda \)) of light capable of breaking the C–C bond can be expressed in SI unit as:
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Longest wavelength corresponds to minimum energy required to break a bond.
To determine the longest wavelength \( \lambda \) of light capable of breaking the C–C bond, we need to analyze the information given. The heat of atomization of methane \((CH_4)\) is \( x \) kJ mol\(^{-1}\), and for ethane \((C_2H_6)\) it is \( y \) kJ mol\(^{-1}\).
Firstly, we express the breaking of a C-C bond in terms of energy and consider the following reactions:
For Methane \(CH_4 \rightarrow C + 4H\): The heat of atomization is \( x \) kJ mol\(^{-1}\), which breaks four C-H bonds.
For Ethane \(C_2H_6 \rightarrow 2C + 6H\): The heat of atomization is \( y \) kJ mol\(^{-1}\), which breaks six C-H bonds and one C-C bond.
The energy required to break six C-H bonds is \( 6x \), and for one C-C bond in ethane is thus \((y - 6x)\) kJ mol\(^{-1}\).
The energy \( E \) required to break a bond can be related to the wavelength \( \lambda \) of light via the equation:
\(E = \dfrac{hc}{\lambda}\)
Where:
\(h\) is Planck's constant (\(6.626 \times 10^{-34}\) J·s).
\(c\) is the speed of light (\(3 \times 10^8\) m/s).
\(\lambda\) is the wavelength in meters.
Thus, rearranging for \(\lambda\):
\(\lambda = \dfrac{hc}{E}\)
For 1 mole of bonds broken, multiply \(E\) (in J) by Avogadro's number \(N_A = 6.022 \times 10^{23}\) mol\(^{-1}\):
The energy for breaking the C-C bond, expressed in joules, is \( (y - 6x) \times 1000 \) J mol\(^{-1}\).
