To calculate the bond length of a carbon-hydrogen bond in ethane, we can use the geometry of the molecule and the law of cosines. Ethane has a tetrahedral geometry around each carbon atom, with a bond angle of 109.5 between the carbon-hydrogen bonds.Let's consider a triangle formed by one carbon atom, one hydrogen atom, and the midpoint of the carbon-carbon bond. The bond angle between the carbon-hydrogen bond and the line connecting the midpoint of the carbon-carbon bond to the hydrogen atom is half of the bond angle between carbon and hydrogen, which is 109.5/2 = 54.75.Let C be the carbon atom, H be the hydrogen atom, and M be the midpoint of the carbon-carbon bond. We are given the distance between carbon atoms CM as 1.54 /2 = 0.77 since M is the midpoint . We want to find the bond length CH.Using the law of cosines, we have:CH^2 = CM^2 + MH^2 - 2 * CM * MH * cos 54.75 Let x be the bond length CH, which is equal to MH. Then:x^2 = 0.77^2 + x^2 - 2 * 0.77 * x * cos 54.75 Now, we can solve for x:x^2 - 2 * 0.77 * x * cos 54.75 = 0.77^2x 1.09 So, the bond length of a carbon-hydrogen bond in ethane is approximately 1.09 .