General considerations

When V changes to V +dV at constant temperature, U changes to

The coefficient (∂U/∂V) T, the slope of a plot of U against V at constant temperature, is the partial derivative of U with respect to V (Fig. 2.21). If, instead, T changes to T + dT at constant volume (Fig. 2.22), then the internal energy changes to

Fig. 2.21 The partial derivative (∂U/∂V) _T is the slope of U with respect to V with the temperature T held constant.

Fig. 2.22 The partial derivative (∂U/∂T) _V is the slope of U with respect to T with the volume V held constant.

Now suppose that V and T both change infinitesimally (Fig. 2.23). The new internal energy, neglecting second-order infinitesimals (those proportional to dV dT), is the sum of the changes arising from each increment:

As a result of the infinitesimal changes in conditions, the internal energy U′ differs from U by the infinitesimal amount dU, so we an write U′=U + dU. Therefore, from the last equation we obtain the very important result that

The interpretation of this equation is that, in a closed system of constant composition, any infinitesimal change in the internal energy is proportional to the infinitesimal changes of volume and temperature, the coefficients of proportionality being the two partial derivatives. In many cases partial derivatives have a straightforward physical interpretation, and thermodynamics gets shapeless and difficult only when that interpretation is not kept in sight. In the present case, we have already met (∂U/∂T) _V in eqn 2.15, where we saw that it is the constant-volume heat capacity, CV. The other coefficient, (∂U/∂V) _T, plays a major role in thermodynamics because it is a measure of the variation of the internal energy of a substance as its volume is changed at constant temperature (Fig. 2.24). We shall denote it πT and, because it has the same dimensions as pressure, call it the internal pressure:

In terms of the notation CV and πT, eqn 2.40 can now be written

dU=π_TdV +C_VdT

Fig. 2.23 An overall change in U, which is denoted dU, arises when both V and T are allowed to change. If second-order infinitesimals are ignored, the overall change is the sum of changes for each variable separately.

Fig. 2.24 The internal pressure, πT, is the slope of U with respect to V with the temperature T held constant.

اخر الاخبار

اخبار العتبة العباسية المقدسة

العتبة العباسية المقدسة تطلق مسابقتها الثقافية بذكرى ولادة الإمام زين العابدين (عليه السلام)

العتبة العباسية المقدسة تكرّم 1000 طالبة من المرتديات العباءة العراقية في مدارس الديوانية

العتبة العباسية المقدسة تؤكد أهمية الحجاب كجزء من منظومة الواجبات الشرعية

العتبة العباسية المقدسة تقيم مهرجانها السنوي الأوَّل لتكريم الطالبات المرتديات للعباءة العراقية في المدارس

المزيد

الآخبار الصحية

الخبز الأسمر أم الأبيض.. أيهما أفضل لضبط سكر الدم؟

عقاقير التخسيس و"الانتحار".. الحكومة الأميركية تحسم الجدل

الهرم الغذائي الجديد.. لماذا يعد "انقلابا" في خارطة الطعام؟

لحماية ظهرك وقلبك في آن واحد.. ما وضعية النوم الصحيحة؟

المزيد

الآخبار التكنلوجية

كيف يعزز الضوء الطبيعي تفكيرك وانتباهك خلال اليوم

سابقة.. طاقم من محطة الفضاء الدولية يعود إلى الأرض مبكرا

منصات التواصل تضرّ بنفسية المراهقين… والفتيات الأكثر تأثرا

ابتكار ألماني قد يغيّر مستقبل صناعة البلاستيك

المزيد

مواضيع ذات صلة

Units

Names of quantities

Working galvanic cells

Electrolysis

Experimental techniques

The rate laws

The electric potential at the interface

The structure of the interface

Catalytic activity at surfaces

Mechanisms of heterogeneous catalysis

Mobility on surfaces

The rate of desorption