Cronbach's Alpha for a new test. tested w control (n=15) and clinical group (n=15), should I run it on whole sample (n=30)or separately on each group?

#1
We have developed a novel test of theory of mind which we administered to a group of controls (n=15) and a group of individuals with motor neurone disease (MND) (n=15). The group with MND performed significantly poorer on our novel test of theory of mind compared to the controls (with a large effect size).

Following my viva, I have been asked to do an internal validity analysis using Cronbach's alpha.

Qu1. Can anyone advise should this be done on the whole sample n=30 (mixing controls and MND patients)?

Or should I run two separate Cronbach's tests one for the control groups responses on our theory of mind test (n=15) and one on our clinical groups responses on the theory of mind test (n=15)

Qu2. Will a small sample size reduce the power of the Cronbach's alpha- as in may the small sample size explain any Cronbach's alphas of less than 0.7 which I understand to be the "acceptable" value for internal validity?

Thank you in advance for any help with this!
Iona