die Eigenschaft
characteristic
A typical feature or character trait that distinguishes a person, object, or substance.
Geduld ist eine sehr nützliche Eigenschaft.
Patience is a very useful characteristic.

characteristic / capacity
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characteristic
A typical feature or character trait that distinguishes a person, object, or substance.
Geduld ist eine sehr nützliche Eigenschaft.
Patience is a very useful characteristic.

The word is made up of 'eigen' (own) and the suffix '-schaft' (-ship/-ness). It therefore describes something that is 'eigen' (inherent) to a person or object, meaning that it belongs to them or it.
Ehrlichkeit ist eine wichtige Eigenschaft für eine Freundschaft.
Honesty is an important characteristic for a friendship.
Common verbs are 'eine Eigenschaft haben' (to have a characteristic) or 'besitzen' (to possess). Adjectives that often come before it are 'positiv' (positive), 'negativ' (negative), 'gut' (good), 'schlecht' (bad), or 'charakteristisch' (characteristic).
Dieses Material hat die Eigenschaft, sehr leicht und trotzdem stabil zu sein.
This material has the characteristic of being very light and yet stable.
This meaning is used almost exclusively in the fixed expression 'in seiner/ihrer Eigenschaft als ...' (in his/her capacity as ...). It expresses that someone is acting in a particular official role or function.
In ihrer Eigenschaft als Ärztin riet sie ihm, sich auszuruhen.
In her capacity as a doctor, she advised him to rest.
The expression 'in seiner/ihrer Eigenschaft als' (in his/her capacity as) is very formal. You mainly find it in written language, in official speeches, or in legal contexts.
Der Bürgermeister begrüßte die Gäste in seiner Eigenschaft als Gastgeber.
The mayor welcomed the guests in his capacity as host.
role / roll
'die Rolle' and 'die Eigenschaft' can both have to do with a person's function. 'Die Rolle' is the normal word when describing what task or importance someone has in a situation. 'In seiner or ihrer Eigenschaft als' also means that someone is acting in a particular function, but it sounds much more formal. Learners may confuse the words because 'Eigenschaft' otherwise usually means a characteristic, for example patience or honesty. For everyday language and normal conversations, 'Rolle' usually fits best; the expression with 'Eigenschaft' is more often found in official texts or formal situations.

In seiner Eigenschaft als Arzt darf Herr Kaya diese Bescheinigung unterschreiben.
In his capacity as a doctor, Mr. Kaya is allowed to sign this certificate.
Im Projekt übernimmt Lena die Rolle der Teamleiterin.
In the project, Lena takes on the role of team leader.